This book gives an excellent and up-to-date overview on the convergence and joint progress in the fields of Generalized Functions and Fourier Analysis, notably in the core disciplines of pseudodifferential operators, microlocal analysis and time-frequency analysis. The volume is a collection of chapters addressing these fields, their interaction, their unifying concepts and their applications and is based on scientific activities related to the International Association for Generalized Functions (IAGF) and the ISAAC interest groups on Pseudo-Differential Operators (IGPDO) and on Generalized Functions (IGGF), notably on the longstanding collaboration of these groups within ISAAC.

Over the past few years a certain shift of focus within the theory of algebras of generalized functions (in the sense of J. F. Colombeau) has taken place. Originating in infinite dimensional analysis and initially applied mainly to problems in nonlinear partial differential equations involving singularities, the theory has undergone a change both in in ternal structure and scope of applicability, due to a growing number of applications to questions of a more geometric nature. The present book is intended to provide an in-depth presentation of these develop ments comprising its structural aspects within the theory of generalized functions as well as a (selective but, as we hope, representative) set of applications. This main purpose of the book is accompanied by a number of sub ordinate goals which we were aiming at when arranging the material included here. First, despite the fact that by now several excellent mono graphs on Colombeau algebras are available, we have decided to give a self-contained introduction to the field in Chapter 1. Our motivation for this decision derives from two main features of our approach. On the one hand, in contrast to other treatments of the subject we base our intro duction to the field on the so-called special variant of the algebras, which makes many of the fundamental ideas of the field particularly transpar ent and at the same time facilitates and motivates the introduction of the more involved concepts treated later in the chapter."

This volumeaddressestheissueofuncertaintyincivilengineeringfromdesign toconstruction. Failures do occur in practice. Attributing them to a residual risk or a faulty execution of the projectdoesnotproperlycover the range of causes. A closer scrutiny of the design, the engineering model, the data, the soil-structure-interactionand the model assumptions is required. Usually, the uncertaintiesininitialandboundaryconditionsaswellasmaterialparameters are abundant. Current engineering practice often leaves these issues aside, despite the factthatnewscienti?c tools have been developed in the past decades that allow a rational description of uncertainties of all kinds, from model uncertainty to data uncertainty. It is the aim of this volume to have a critical look atcurrent engineering riskconcepts in order to raise awareness of uncertainty in numericalcom- tations, shortcomings of a strictly probabilistic safety concept, geotechnical models of failure mechanisms and their implications forconstruction mana- ment, execution, and the juristic questionas to who has to takeresponsibility. In addition, a number ofthe new procedures for modelling uncertaintyare- plained. Our central claim is that doubts and uncertainties must be openly - dressed in the design process. This contrasts certain tendencies in the en- neering community that, though incorporating uncertainties by one or the other way in the modelling process, claim to being able tocontrol the

This volumeaddressestheissueofuncertaintyincivilengineeringfromdesign toconstruction. Failures do occur in practice. Attributing them to a residual risk or a faulty execution of the projectdoesnotproperlycover the range of causes. A closer scrutiny of the design, the engineering model, the data, the soil-structure-interactionand the model assumptions is required. Usually, the uncertaintiesininitialandboundaryconditionsaswellasmaterialparameters are abundant. Current engineering practice often leaves these issues aside, despite the factthatnewscienti?c tools have been developed in the past decades that allow a rational description of uncertainties of all kinds, from model uncertainty to data uncertainty. It is the aim of this volume to have a critical look atcurrent engineering riskconcepts in order to raise awareness of uncertainty in numericalcom- tations, shortcomings of a strictly probabilistic safety concept, geotechnical models of failure mechanisms and their implications forconstruction mana- ment, execution, and the juristic questionas to who has to takeresponsibility. In addition, a number ofthe new procedures for modelling uncertaintyare- plained. Our central claim is that doubts and uncertainties must be openly - dressed in the design process. This contrasts certain tendencies in the en- neering community that, though incorporating uncertainties by one or the other way in the modelling process, claim to being able tocontrol the

Over the past few years a certain shift of focus within the theory of algebras of generalized functions (in the sense of J. F. Colombeau) has taken place. Originating in infinite dimensional analysis and initially applied mainly to problems in nonlinear partial differential equations involving singularities, the theory has undergone a change both in in ternal structure and scope of applicability, due to a growing number of applications to questions of a more geometric nature. The present book is intended to provide an in-depth presentation of these develop ments comprising its structural aspects within the theory of generalized functions as well as a (selective but, as we hope, representative) set of applications. This main purpose of the book is accompanied by a number of sub ordinate goals which we were aiming at when arranging the material included here. First, despite the fact that by now several excellent mono graphs on Colombeau algebras are available, we have decided to give a self-contained introduction to the field in Chapter 1. Our motivation for this decision derives from two main features of our approach. On the one hand, in contrast to other treatments of the subject we base our intro duction to the field on the so-called special variant of the algebras, which makes many of the fundamental ideas of the field particularly transpar ent and at the same time facilitates and motivates the introduction of the more involved concepts treated later in the chapter."

This book gives an excellent and up-to-date overview on the convergence and joint progress in the fields of Generalized Functions and Fourier Analysis, notably in the core disciplines of pseudodifferential operators, microlocal analysis and time-frequency analysis. The volume is a collection of chapters addressing these fields, their interaction, their unifying concepts and their applications and is based on scientific activities related to the International Association for Generalized Functions (IAGF) and the ISAAC interest groups on Pseudo-Differential Operators (IGPDO) and on Generalized Functions (IGGF), notably on the longstanding collaboration of these groups within ISAAC.

This easy-to-follow textbook/reference presents a concise introduction to mathematical analysis from an algorithmic point of view, with a particular focus on applications of analysis and aspects of mathematical modelling. The text describes the mathematical theory alongside the basic concepts and methods of numerical analysis, enriched by computer experiments using MATLAB, Python, Maple, and Java applets. This fully updated and expanded new edition also features an even greater number of programming exercises. Topics and features: describes the fundamental concepts in analysis, covering real and complex numbers, trigonometry, sequences and series, functions, derivatives, integrals, and curves; discusses important applications and advanced topics, such as fractals and L-systems, numerical integration, linear regression, and differential equations; presents tools from vector and matrix algebra in the appendices, together with further information on continuity; includes added material on hyperbolic functions, curves and surfaces in space, second-order differential equations, and the pendulum equation (NEW); contains experiments, exercises, definitions, and propositions throughout the text; supplies programming examples in Python, in addition to MATLAB (NEW); provides supplementary resources at an associated website, including Java applets, code source files, and links to interactive online learning material. Addressing the core needs of computer science students and researchers, this clearly written textbook is an essential resource for undergraduate-level courses on numerical analysis, and an ideal self-study tool for professionals seeking to enhance their analysis skills.

Diese grundlegende Einfuhrung in die Analysis wendet sich an Informatiker im ersten Studienabschnitt. Um speziell auf die Bedurfnisse des Informatikstudiums einzugehen, haben die Autoren diesem Werk folgende Konzepte zugrunde gelegt: Algorithmischer Zugang, schlanke Darstellung, Software als integrativer Bestandteil, Betonung von Modellbildung und Anwendungen der Analysis. Der Gegenstand des Buches liegt im Spannungsfeld zwischen Mathematik, Informatik und Anwendungen. Hier kommt dem algorithmischen Denken ein hoher Stellenwert zu. Der gewahlte Zugang beinhaltet: Entwicklung der Grundlagen der Analysis aus algorithmischer Sichtweise, Vergegenstandlichung der Theorie mittels MATLAB- und Maple-Programmen und Java-Applets, Behandlung grundlegender Konzepte und Verfahren der numerischen Analysis. Das Buch kann ab dem ersten Semester als Vorlesungsgrundlage, als Begleittext zu einer Vorlesung oder im Selbststudium verwendet werden."