Regning og matematisk analyse

Basierend auf Grundkenntnissen aus der Schulzeit oder aus dem ersten Band des Gesamtwerks ¿Mathematik verstehen und anwenden¿ führt dieser zweite Band in die Vektoranalysis, in das Gebiet der Differenzialgleichungen und in die Fourier-Analysis einschließlich der Laplace-Transformation ein und beinhaltet außerdem eine Einführung in die Wahrscheinlichkeitsrechnung und Statistik. Damit er unabhängig vom ersten Band gelesen werden kann, beginnt er mit einer kurzen Zusammenfassung der wichtigsten Begriffe und Ergebnisse der Differenzial- und Integralrechnung sowie der Linearen Algebra.Zielgruppe sind Studierende der Ingenieur- und Naturwissenschaften an Fachhochschulen und Universitäten. Trotz der verständlichen Darstellung für ein Bachelor-Studium geht die mathematische Exaktheit nicht verloren. Hintergrundinformationen und Beweise ergänzen die sehr umfangreiche Stoffauswahl und bieten Anknüpfungspunkte für ein Masterstudium. Daneben erleichtern sie auch den Einstieg in Spezialvorlesungen der Mathematik wie beispielsweise die Numerik, die Funktionalanalysis und insbesondere die Fourier-Analysis.In der vierten Auflage wurden viele Anwendungsbeispiele ergänzt und der Text grundlegend überarbeitet.Stimmen zur ersten Auflage:¿Sowohl mathematisch exakt als auch äußerst anschaulich. Eine echte Bereicherung der großen Auswahl an Büchern zum Thema Ingenieurmathematik.¿Prof. Dr. Andreas Gessinger, Rheinische Fachhochschule Köln¿Der Spagat zwischen Verständlichkeit und mathematischer Tiefe ist hervorragend gelungen. Eine breite Palette von praxisorientierten Beispielen wirkt motivationsfördernd.¿Prof. Dr. Helga Tecklenburg, Hochschule für Technik, Wirtschaft und Kultur Leipzig

Dieses Lehrbuch beruht auf fast 20 Jahren Erfahrung aus Vorlesungen und Examenskursen. Es richtet sich an Bachelor-/Master-Studierende wie an Studierende des gymnasialen Lehramts und ist speziell auf letztere zugeschnitten: Der Inhalt orientiert sich an den Anforderungen des bayerischen Staatsexamens; er wird durch 40 Examensaufgaben mit detaillierten Lösungen illustriert. Die ¿ von einer Vielzahl von Grafiken veranschaulichte ¿ Darstellung legt großen Wert auf möglichst widerstandsarme Lesbarkeit, ausführliche Erläuterungen, gründliche Motivationen und das Aufzeigen von Zusammenhängen. Zahlreiche (teils auch für die Oberstufe geeignete) Beispiele beleuchten die überragende Bedeutung von Differentialgleichungen in Natur, Wissenschaft und Technik.Inhaltlich liegt ein gewisser Fokus auf der qualitativen Theorie der Differentialgleichungen (insbesondere der Stabilitätstheorie), deren Ziel es ist, das Verhalten von Lösungen auch dann zu ¿verstehen¿, wenn man sie nicht explizit angeben kann.

Gegen Angst vor Mathematik hilft Verstehen. Dieses Buch setzt nur elementare Schulkenntnisse voraus und entwickelt die Mathematik schrittweise und systematisch von der Bruchrechnung über die Differenzial- und Integralrechnung sowie die Lineare Algebra bis zur Funktionalanalysis. Dabei werden Sie vom Vertrauten zum Neuen geführt. Neben vielen Anwendungsbeispielen aus den Ingenieurwissenschaften finden Sie zu jedem Teil des Buchs zahlreiche Aufgaben (mit Lösungen im elektronischen Zusatzmaterial).Mit der hier vorliegenden vierten Auflage wurde der Text grundlegend überarbeitet, es wurden neue Anwendungsbeispiele aufgenommen, und das Gesamtwerk ¿Mathematik verstehen und anwenden¿ wurde aufgrund des Umfangs in zwei Bände aufgeteilt. Dieser erste Band wird durch einen zweiten ergänzt, in dem weiterführende Themen wie die Vektoranalysis, Differenzialgleichungen und die Fourier-Analysis behandelt werden. Außerdem beinhaltet der zweite Band eine Einführung in die Wahrscheinlichkeitsrechnung und Statistik.Zielgruppe sind Studierende der Ingenieur- und Naturwissenschaften an Fachhochschulen und Universitäten. Trotz der verständlichen Darstellung für ein Bachelor-Studium geht die mathematische Exaktheit nicht verloren. Hintergrundinformationen und Beweise ergänzen die sehr umfangreiche Stoffauswahl und bieten Anknüpfungspunkte für ein Masterstudium. Daneben erleichtern sie auch den Einstieg in Spezialvorlesungen der Mathematik wie beispielsweise die Numerik, die Funktionalanalysis und insbesondere die Fourier-Analysis.Stimmen zur ersten Auflage:¿Sowohl mathematisch exakt als auch äußerst anschaulich. Eine echte Bereicherung der großen Auswahl an Büchern zum Thema Ingenieurmathematik.¿Prof. Dr. Andreas Gessinger, Rheinische Fachhochschule Köln¿Der Spagat zwischen Verständlichkeit und mathematischer Tiefe ist hervorragend gelungen. Eine breite Palette von praxisorientierten Beispielen wirkt motivationsfördernd.¿Prof. Dr. Helga Tecklenburg, Hochschule für Technik, Wirtschaft und Kultur Leipzig

This graduate textbook provides a self-contained introduction to the classical theory of partial differential equations (PDEs). Through its careful selection of topics and engaging tone, readers will also learn how PDEs connect to cutting-edge research and the modeling of physical phenomena. The scope of the Third Edition greatly expands on that of the previous editions by including five new chapters covering additional PDE topics relevant for current areas of active research. This includes coverage of linear parabolic equations with measurable coefficients, parabolic DeGiorgi classes, Navier-Stokes equations, and more. The ¿Problems and Complements¿ sections have also been updated to feature new exercises, examples, and hints toward solutions, making this a timely resource for students.Partial Differential Equations: Third Edition is ideal for graduate students interested in exploring the theory of PDEs and how they connect to contemporary research. It can also serve as a useful tool for more experienced readers who are looking for a comprehensive reference.

This book includes a thorough theoretical and computational analysis of unconstrained and constrained optimization algorithms and combines and integrates the most recent techniques and advanced computational linear algebra methods. Nonlinear optimization methods and techniques have reached their maturity and an abundance of optimization algorithms are available for which both the convergence properties and the numerical performances are known. This clear, friendly, and rigorous exposition discusses the theory behind the nonlinear optimization algorithms for understanding their properties and their convergence, enabling the reader to prove the convergence of his/her own algorithms. It covers cases and computational performances of the most known modern nonlinear optimization algorithms that solve collections of unconstrained and constrained optimization test problems with different structures, complexities, as well as those with large-scale real applications. The book is addressed to all those interested in developing and using new advanced techniques for solving large-scale unconstrained or constrained complex optimization problems. Mathematical programming researchers, theoreticians and practitioners in operations research, practitioners in engineering and industry researchers, as well as graduate students in mathematics, Ph.D. and master in mathematical programming will find plenty of recent information and practical approaches for solving real large-scale optimization problems and applications.

This book provides the rigorous mathematical foundations of Quantum Physics, from the operational meaning of the measuring process to the most recent theories for the quantum scale of space-time geometry. Topics like relativistic invariance, quantum systems with finite and infinitely many degrees of freedom, second quantisation, scattering theory, are all presented through the formalism of Operator Algebras for a precise mathematical justification.The book is targeted to graduate students and researchers in the area of theoretical/mathematical physics who want to learn about the mathematical foundations of quantum physics, as well as the mathematics students and researchers in the area of operator algebras/functional analysis who want to dive into some of the applications of the theory to physics. 

This textbook describes selected topics in functional analysis as powerful tools of immediate use in many fields within applied mathematics, physics and engineering. It follows a very reader-friendly structure, with the presentation and the level of exposition especially tailored to those who need functional analysis but don't have a strong background in this branch of mathematics. For every tool, this work emphasizes the motivation, the justification for the choices made, and the right way to employ the techniques. Proofs appear only when necessary for the safe use of the results. The book gently starts with a road map to guide reading. A subsequent chapter recalls definitions and notation for abstract spaces and some function spaces, while Chapter 3 enters dual spaces. Tools from Chapters 2 and 3 find use in Chapter 4, which introduces distributions. The Linear Functional Analysis basic triplet makes up Chapter 5, followed by Chapter 6, which introduces the concept of compactness. Chapter 7 brings a generalization of the concept of derivative for functions defined in normed spaces, while Chapter 8 discusses basic results about Hilbert spaces that are paramount to numerical approximations. The last chapter brings remarks to recent bibliographical items. Elementary examples included throughout the chapters foster understanding and self-study. By making key, complex topics more accessible, this book serves as a valuable resource for researchers, students, and practitioners alike that need to rely on solid functional analysis but don't need to delve deep into the underlying theory.

This book gathers, in a beautifully structured way, recent findings on chain conditions in commutative algebra that were previously only available in papers. The majority of chapters are self-contained, and all include detailed proofs, a wealth of examples and solved exercises, and a complete reference list. The topics covered include S-Noetherian, S-Artinian, Nonnil-Noetherian, and Strongly Hopfian properties on commutative rings and their transfer to extensions such as polynomial and power series rings, and more. Though primarily intended for readers with a background in commutative rings, modules, polynomials and power series extension rings, the book can also be used as a reference guide to support graduate-level algebra courses, or as a starting point for further research.

This volume gathers select proceedings of the 10th International Conference on Wave Mechanics and Vibrations (WMVC), held in Lisbon, Portugal, on July 4-6, 2022. It covers recent developments and cutting-edge methods in wave mechanics and vibrations applied to a wide range of engineering problems. It presents analytical and computational studies in structural mechanics, seismology and earthquake engineering, mechanical engineering, aeronautics, robotics and nuclear engineering among others. The volume will be of interest for students, researchers, and professionals interested in the wide-ranging applications of wave mechanics and vibrations. 

This proceedings volume gathers selected, peer-reviewed papers presented at the 41st International Conference on Infinite Dimensional Analysis, Quantum Probability and Related Topics (QP41) that was virtually held at the United Arab Emirates University (UAEU) in Al Ain, Abu Dhabi, from March 28th to April 1st, 2021. The works cover recent developments in quantum probability and infinite dimensional analysis, with a special focus on applications to mathematical physics and quantum information theory. Covered topics include white noise theory, quantum field theory, quantum Markov processes, free probability, interacting Fock spaces, and more. By emphasizing the interconnection and interdependence of such research topics and their real-life applications, this reputed conference has set itself as a distinguished forum to communicate and discuss new findings in truly relevant aspects of theoretical and applied mathematics, notably in the field of mathematical physics, as well as an event of choice for the promotion of mathematical applications that address the most relevant problems found in industry. That makes this volume a suitable reading not only for researchers and graduate students with an interest in the field but for practitioners as well.

Dieses Buch ist Teil eines 6-bändigen Werks. Es befasst sich mit der Anwendung von Differentialgleichungen in diversen Bereichen der Physik und dem Ingenieurwesen. Dabei wird die Bilanzgleichung ins Zentrum der Betrachtung gerückt. Die Lesenden lernen Schritt für Schritt, wie ein konkret gestelltes Problem mit Hilfe sinnvoller Voraussetzungen und Idealisierungen modelliert, als Bilanz formuliert und formalisiert und die entstandene Differentialgleichung bzw. das entstandene Differentialgleichungssystem exakt oder numerisch gelöst wird. Dieses didaktische Konzept wird durchgehend sorgfältig und konsequent für jedes Teilgebiet angewandt und ermöglicht auf diese Weise den Studierenden die Bilanz als etwas Grundlegendes zur Beschreibung eines physikalischen Sachverhalts zu begreifen. Sämtliche theoretisch gewonnenen Ergebnisse werden in Worten festgehalten, teilweise mit Hilfe von Computersimulationen dargestellt und von vielen, konkreten, vollständig gelösten Beispielen begleitet.

Dieses Buch ist Teil eines 6-bändigen Werks. Es ist als Wegweiser für den Einstieg in die Modellierung von Populationen und deren Interaktion gedacht. Die Lesenden erfahren in nachvollziehbaren Schritten, wie ein Ökosystem unter sinnvollen Annahmen und Idealisierungen mit Hilfe von Differentialgleichungen modelliert werden kann und wie die bestehenden Modelle bei einem Eingriff des Menschen oder aufgrund der Berücksichtigung von Phänomenen wie etwa intra- und interspezifische Konkurrenz oder Begrenzung des Nahrungsangebots usw. angepasst werden müssen. Die dabei entstandene Differentialgleichung bzw. das entstandene Differentialgleichungssystem wird exakt oder numerisch gelöst, die Ergebnisse in Worten und mit Hilfe von Computersimulationen festgehalten und von vielen, konkreten, vollständig gelösten Beispielen begleitet.

Dieses 6-bändige Werk befasst sich mit der Anwendung von Differentialgleichungen in diversen Bereichen der Physik und dem Ingenieurwesen. Dabei wird die Bilanzgleichung ins Zentrum der Betrachtung gerückt. Die Lesenden lernen Schritt für Schritt, wie ein konkret gestelltes Problem mit Hilfe sinnvoller Voraussetzungen und Idealisierungen modelliert, als Bilanz formuliert und formalisiert und die entstandene Differentialgleichung bzw. das entstandene Differentialgleichungssystem exakt oder numerisch gelöst wird. Dieses didaktische Konzept wird durchgehend sorgfältig und konsequent für jedes Teilgebiet angewandt und ermöglicht auf diese Weise den Studierenden die Bilanz als etwas Grundlegendes zur Beschreibung eines physikalischen Sachverhalts zu begreifen. Sämtliche theoretisch gewonnenen Ergebnisse werden in Worten festgehalten und von vielen, konkreten, vollständig gelösten Beispielen begleitet.

This volume highlights contributions of women mathematicians in the study of complex materials and includes both original research papers and reviews. The featured topics and methods draw on the fields of Calculus of Variations, Partial Differential Equations, Functional Analysis, Differential Geometry and Topology, as well as Numerical Analysis and Mathematical Modelling. Areas of applications include foams, fluid-solid interactions, liquid crystals, shape-memory alloys, magnetic suspensions, failure in solids, plasticity, viscoelasticity, homogenization, crystallization, grain growth, and phase-field models.

This monograph provides a state-of-the-art, self-contained account on the effectiveness of the method of boundary layer potentials in the study of elliptic boundary value problems with boundary data in a multitude of function spaces. Many significant new results are explored in detail, with complete proofs, emphasizing and elaborating on the link between the geometric measure-theoretic features of an underlying surface and the functional analytic properties of singular integral operators defined on it. Graduate students, researchers, and professionals interested in a modern account of the topic of singular integral operators and boundary value problems - as well as those more generally interested in harmonic analysis, PDEs, and geometric analysis - will find this text to be a valuable addition to the mathematical literature.

In addition to expanding and clarifying a number of sections of the first edition, it generalizes the analysis that eliminates the noncausal pre-acceleration so that it applies to removing any pre-deceleration as well. It also introduces a robust power series solution to the equation of motion that produces an extremely accurate solution to problems such as the motion of electrons in uniform magnetic fields.

This book offers a primer on the fundamentals and applications of the Geroch-Held-Penrose (GHP) calculus, a powerful formalism designed for spacetimes that occur frequently in the teaching of General Relativity. Specifically, the book shows in detail the power of the calculus when dealing with spherically symmetric spacetimes. After introducing the basics, a new look at all the classical spherically symmetric black hole solutions is given within the GHP formalism. This is then employed to give new insights into the Tolman-Oppenheimer-Volkoff equations for stellar structure, including a derivation of new exact anisotropic fluid solutions. Finally, a re-writing of some essential features of black hole thermodynamics within the GHP formalism is performed. The book is based on the authors' lecture notes, used in their undergraduate and graduate lectures and while supervising their upper undergraduate and graduate students. To fully benefit from this concise primer, readers only need an undergraduate background in general relativity.

This book systematically establishes the almost periodic theory of dynamic equations and presents applications on time scales in fuzzy mathematics and uncertainty theory. The authors introduce a new division of fuzzy vectors depending on a determinant algorithm and develop a theory of almost periodic fuzzy multidimensional dynamic systems on time scales. Several applications are studied; in particular, a new type of fuzzy dynamic systems called fuzzy q-dynamic systems (i.e. fuzzy quantum dynamic systems) is presented. The results are not only effective on classical fuzzy dynamic systems, including their continuous and discrete situations, but are also valid for other fuzzy multidimensional dynamic systems on various hybrid domains. In an effort to achieve more accurate analysis in real world applications, the authors propose a number of uncertain factors in the theory. As such, fuzzy dynamical models, interval-valued functions, differential equations, fuzzy-valued differential equations, and their applications to dynamic equations on time scales are considered.

This monograph examines the stability of various coupled systems with local Kelvin-Voigt damping. The development of this area is thoroughly reviewed along with the authors' contributions. New results are featured on the fundamental properties of solutions of linear transmission evolution PDEs involving Kelvin-Voigt damping, with special emphasis on the asymptotic behavior of these solutions. The vibrations of transmission problems are highlighted as well, making this a valuable resource for those studying this active area of research. The book begins with a brief description of the abstract theory of linear evolution equations with a particular focus on semigroup theory. Different types of stability are also introduced along with their connection to resolvent estimates. After this foundation is established, different models are presented for uni-dimensional and multi-dimensional linear transmission evolution partial differential equations with Kelvin-Voigt damping. Stabilization of Kelvin-Voigt Damped Systems will be a useful reference for researchers in mechanics, particularly those interested in the study of control theory of PDEs.

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 volume constitutes the thoroughly refereed proceedings of the 49th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2023. The 33 full papers presented in this volume were carefully reviewed and selected from a total of 116 submissions. The WG 2022 workshop aims to merge theory and practice by demonstrating how concepts from graph theory can be applied to various areas in computer science, or by extracting new graph theoretic problems from applications.

This volume LNCS 14240 constitutes the refereed proceedings of the 30th International Symposium on String Processing and Information Retrieval, SPIRE 2023, held in Pisa, Italy, during September 26¿28, 2023. The 31 full papers presented were carefully reviewed and selected from 47 submissions. They cover topics such as: data structures; algorithms; constrained Substring complexity; data compression codes; succinct k-spectra; and LCP array of wheeler DFAs.

This book introduces the basic concept of a dissipative soliton, before going to explore recent theoretical and experimental results for various classes of dissipative optical solitons, high-energy dissipative solitons and their applications, and mode-locked fiber lasers.A soliton is a concept which describes various physical phenomena ranging from solitary waves forming on water to ultrashort optical pulses propagating in an optical fiber. While solitons are usually attributed to integrability, in recent years the notion of a soliton has been extended to various systems which are not necessarily integrable.  Until now, the main emphasis has been given to well-known conservative soliton systems, but new avenues of inquiry were opened when physicists realized that solitary waves did indeed exist in a wide range of non-integrable and non-conservative systems leading to the concept of so-called dissipative optical solitons.Dissipative optical solitons have many unique properties which differ from those of their conservative counterparts. For example, except for very few cases, they form zero-parameter families and their properties are completely determined by the external parameters of the optical system. They can exist indefinitely in time, as long as these parameters stay constant. These features of dissipative solitons are highly desirable for several applications, such as in-line regeneration of optical data streams and generation of stable trains of laser pulses by mode-locked cavities.

The recent book of the series continues the collection of articles dealing with the important and efficient combination of traditional and novel mathematical approaches with various computational intelligence techniques, with a stress of fuzzy systems, and fuzzy logic.Complex systems are theoretically intractable, as the need of time and space resources (e.g., computer capacity) exceed any implementable extent. How is it possible that in the practice, such problems are usually manageable with an acceptable quality by human experts? They apply expert domain knowledge and various methods of approximate modeling and corresponding algorithms. Computational intelligence is the mathematical tool box that collects techniques which are able to model such human interaction, while (new) mathematical approaches are developed and used everywhere where the complexity of the sub-task allows it. The innovative approaches in this book give answer to many questions on how to solve "e;unsolvable"e; problems.

This book surveys the foundations of the theory of slice regular functions over the quaternions, introduced in 2006, and gives an overview of its generalizations and applications.As in the case of other interesting quaternionic function theories, the original motivations were the richness of the theory of holomorphic functions of one complex variable and the fact that quaternions form the only associative real division algebra with a finite dimension n>2. (Slice) regular functions quickly showed particularly appealing features and developed into a full-fledged theory, while finding applications to outstanding problems from other areas of mathematics. For instance, this class of functions includes polynomials and power series. The nature of the zero sets of regular functions is particularly interesting and strictly linked to an articulate algebraic structure, which allows several types of series expansion and the study of singularities. Integral representation formulas enrich the theory and are fundamental to the construction of a noncommutative functional calculus. Regular functions have a particularly nice differential topology and are useful tools for the construction and classification of quaternionic orthogonal complex structures, where they compensate for the scarcity of conformal maps in dimension four.This second, expanded edition additionally covers a new branch of the theory: the study of regular functions whose domains are not axially symmetric. The volume is intended for graduate students and researchers in complex or hypercomplex analysis and geometry, function theory, and functional analysis in general.

Das Buch vermittelt Lösungsverfahren für partielle Riccati-Differenzialgleichungen (DGL). Diese DGL appliziert man zur mathematischen Beschreibung der Funktion von Faraday-Effekt-Stromsensoren. Ausgehend vom allgemeinen Lösungsverhalten erfolgt die Definition des individuellen sowie kollektiven Lösungsverhaltens einer beliebigen Gleichung. Reiner Thiele erklärt den Unterschied zwischen Individual- und Partiallösungen. Außerdem kreiert er Simulationsdiagramme und zeigt, dass es sich bei der Hardware um Regelkreise handelt.