Regning og matematisk analyse

Die 3., ergänzte Auflage stellt auf breiter fachlicher Ebene einfache elementare zahlentheoretische Inhalte vor sowie Stoffkomplexe aus der analytischen und algebraischen Zahlentheorie. Das Lehrbuch bietet auf überschaubaren mathematischen Niveau einen leicht verständlichen Einstieg in ausgewählte Themen der Zahlentheorie und beschreibt aktuelle Forschungsergebnisse zum RSA-Algorithmus. Sämtliche Kapitel enthalten umfassende Beispiele, Übungsaufgaben mit Lösungen und ausführlich durchgerechnete Beweise, so dass es sich sehr gut zur Prüfungsvorbereitung eignet.

This open access volume explains the foundations of modern solvers for ordinary differential equations (ODEs). Formulating and solving ODEs is an essential part of mathematical modeling and computational science, and numerous solvers are available in commercial and open source software. However, no single ODE solver is the best choice for every single problem, and choosing the right solver requires fundamental insight into how the solvers work. This book will provide exactly that insight, to enable students and researchers to select the right solver for any ODE problem of interest, or implement their own solvers if needed. The presentation is compact and accessible, and focuses on the large and widely used class of solvers known as Runge-Kutta methods. Explicit and implicit methods are motivated and explained, as well as methods for error control and automatic time step selection, and all the solvers are implemented as a class hierarchy in Python.

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 volume presents an accessible overview of mathematical control theory and analysis of PDEs, providing young researchers a snapshot of these active and rapidly developing areas. The chapters are based on two mini-courses and additional talks given at the spring school "e;Trends in PDEs and Related Fields"e; held at the University of Sidi Bel Abbes, Algeria from 8-10 April 2019. In addition to providing an in-depth summary of these two areas, chapters also highlight breakthroughs on more specific topics such as:Sobolev spaces and elliptic boundary value problemsLocal energy solutions of the nonlinear wave equationGeometric control of eigenfunctions of Schrodinger operatorsResearch in PDEs and Related Fields will be a valuable resource to graduate students and more junior members of the research community interested in control theory and analysis of PDEs.

The Most Effective Introduction To CalculusA self-teaching guide for anyone who wants to learn the concepts that precede Calculus in a clear and concise way. It employs a friendly writing tone, many solved and proposed exercises, stories and even jokes; standing apart from the classic academic book.This book covers all the essential topics of Precalculus, including:ArithmeticAlgebraPolynomialsAnalytic GeometryFunctionsThis title is the result of several decades of the author's experience teaching Calculus at university education level. Most important featuresSuitable for students or applicants for Science and Engineering careers.Suitable for self-learners and Science and Technology enthusiasts.Good balance of theory and practice.Theory is assisted by many examples and solved exercises.More than 750 proposed exercises with their respective answers.Passages of mathematical history and references to contemporary cases.Modern mathematical jokes.Assisted by QR codes that link to complementary information on our social media.Book contentChapter 1. Preliminaries:An Overview of Logic and Set TheoryThe Real Number System Radicals and Rational ExponentsSome Topics of AlgebraPolynomial EquationsOrder Axioms and InequationsAbsolute ValueChapter¿¿¿¿¿¿¿¿¿¿¿¿¿¿¿¿¿¿¿¿¿¿¿2. The Cartesian Plane and The Line:The Cartesian PlaneEquation Graphs. Symmetry and TranslationsLines and Linear EquationsChapter 3. The Conics:¿¿¿¿¿¿¿¿¿¿¿¿¿¿¿ IntroductionThe ParabolaThe EllipseThe HyperbolaGeneral 2nd Degree Equation. RotationsChapter 4. Real Functions:¿¿¿¿¿¿¿¿¿Real Functions and their GraphsTrigonometric FunctionsNew Functions from Old FunctionsInverse Functions Inverse Trigonometric FunctionsExponential FunctionsLogarithmic FunctionsFunction Applications

This open access book gives a systematic introduction into the spectral theory of differential operators on metric graphs. Main focus is on the fundamental relations between the spectrum and the geometry of the underlying graph.The book has two central themes: the trace formula and inverse problems.The trace formula is relating the spectrum to the set of periodic orbits and is comparable to the celebrated Selberg and Chazarain-Duistermaat-Guillemin-Melrose trace formulas. Unexpectedly this formula allows one to construct non-trivial crystalline measures and Fourier quasicrystals solving one of the long-standing problems in Fourier analysis. The remarkable story of this mathematical odyssey is presented in the first part of the book.To solve the inverse problem for Schrödinger operators on metric graphs the magnetic boundary control method is introduced. Spectral data depending on the magnetic flux allow one to solve the inverse problem in full generality, this means to reconstruct not only the potential on a given graph, but also the underlying graph itself and the vertex conditions.The book provides an excellent example of recent studies where the interplay between different fields like operator theory, algebraic geometry and number theory, leads to unexpected and sound mathematical results. The book is thought as a graduate course book where every chapter is suitable for a separate lecture and includes problems for home studies. Numerous illuminating examples make it easier to understand new concepts and develop the necessary intuition for further studies.

This book studies the modules arising in Fourier expansions of automorphic forms, namely Fourier term modules on SU(2,1), the smallest rank one Lie group with a non-abelian unipotent subgroup. It considers the ¿abelian¿ Fourier term modules connected to characters of the maximal unipotent subgroups of SU(2,1), and also the ¿non-abelian¿ modules, described via theta functions. A complete description of the submodule structure of all Fourier term modules is given, with a discussion of the consequences for Fourier expansions of automorphic forms, automorphic forms with exponential growth included.These results can be applied to prove a completeness result for Poincaré series in spaces of square integrable automorphic forms.Aimed at researchers and graduate students interested in automorphic forms, harmonic analysis on Lie groups, and number-theoretic topics related to Poincaré series, the book will also serve as a basic reference on spectral expansion with Fourier-Jacobi coefficients. Only a background in Lie groups and their representations is assumed.

Das vorliegende Buch analysiert den zukünftigen Temperaturverlauf der Erdoberfläche mithilfe eines naturwissenschaftlichen Modells und gibt eine Übersicht über die Ursachen und Auswirkungen des Klimawandels. Der Inhalt umfasst außerdem relevante Daten und Fakten zu den Themen Kohlenstoffbudget, Demographie, Gletscher, Permafrost, Artenvielfalt, Mobilität und Extremwetter. Die rasche globale Reduzierung des CO2-Anstiegs auf Netto-Null ist eine dringende Notwendigkeit, um nachfolgenden Generationen ein menschenwürdiges Leben zu ermöglichen. Die Industriestaaten tragen als Hauptverursacher eine moralische Verantwortung, die erwähnte Reduzierung zügig umzusetzen und die Entwicklungsländer in diesem Prozess zu unterstützen. Das Buch richtet sich auch an Personen mit wenig Bezug zur Mathematik.Die Produktfamilie WissensExpress bietet Ihnen Lehr- und Lernbücher in kompakter Form. Die Bücher liefern schnell und verständlich fundiertes Wissen.

This book constitutes the refereed proceedings of the 5th Iberoamerican Conference and 4th Indo-American Conference on Knowledge Graphs and Semantic Web, KGSWC 2023, held jointly in Zaragoza, Spain, during November 13¿15, 2023.The 18 full and 2 short papers presented were carefully reviewed and selected from 50 submissions. They focus on the following topics: knowledge representation; natural language processing/text mining; and machine/deep learning research.

This monograph presents a comprehensive, self-contained, and novel approach to the Divergence Theorem through five progressive volumes. Its ultimate 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. The text is intended for researchers, graduate students, and industry professionals interested in applications of harmonic analysis and geometric measure theory to complex analysis, scattering, and partial differential equations.Volume I establishes a sharp version of the Divergence Theorem (aka Fundamental Theorem of Calculus) which allows for an inclusive class of vector fields whose boundary trace is only assumed to exist in a nontangential pointwise sense.

This graduate textbook provides an alternative to discrete event simulation. It describes how to formulate discrete event systems, how to convert them into Markov chains, and how to calculate their transient and equilibrium probabilities. The most appropriate methods for finding these probabilities are described in some detail, and templates for efficient algorithms are provided. These algorithms can be executed on any laptop, even in cases where the Markov chain has hundreds of thousands of states. This book features the probabilistic interpretation of Gaussian elimination, a concept that unifies many of the topics covered, such as embedded Markov chains and matrix analytic methods.The material provided should aid practitioners significantly to solve their problems. This book also provides an interesting approach to teaching courses of stochastic processes.

This volume proposes an integral approach to studying the geophysics of Earth. It is motivated by a variety of phenomena from nature with deep and direct impacts in our lives. Such events may evolve across a large range of spatial and time scales and may be observed in the ocean, the atmosphere, the volcanic surface as well as underground.The physical laws dictating the evolution of such phenomena lead to the unifying theme of this manuscript, that is, the mathematical and computational modeling of flows and waves. Consequently, the underlying models are given in terms of Partial Differential Equations (PDEs) whose solutions are approximated using numerical methods, thus providing simulations of the aforementioned phenomena, as well as the appropriate geophysical validation and interpretation.

Analysis at Large is dedicated to Jean Bourgain whose research has deeply influenced the mathematics discipline, particularly in analysis and its interconnections with other fields. In this volume, the contributions made by renowned experts present both research and surveys on a wide spectrum of subjects, each of which pay tribute to a true mathematical pioneer. Examples of topics discussed in this book include Bourgain's discretized sum-product theorem, his work in nonlinear dispersive equations, the slicing problem by Bourgain, harmonious sets, the joint spectral radius, equidistribution of affine random walks, Cartan covers and doubling Bernstein type inequalities, a weighted Prekopa-Leindler inequality and sumsets with quasicubes, the fractal uncertainty principle for the Walsh-Fourier transform, the continuous formulation of shallow neural networks as Wasserstein-type gradient flows, logarithmic quantum dynamical bounds for arithmetically defined ergodic Schrodinger operators, polynomial equations in subgroups, trace sets of restricted continued fraction semigroups, exponential sums, twisted multiplicativity and moments, the ternary Goldbach problem, as well as the multiplicative group generated by two primes in Z/QZ.It is hoped that this volume will inspire further research in the areas of analysis treated in this book and also provide direction and guidance for upcoming developments in this essential subject of mathematics.

The intention of this book is to shine a bright light on the intellectual context of Euler's contributions to physics and mathematical astronomy. Leonhard Euler is one of the most important figures in the history of science, a blind genius who introduced mathematical concepts and many analytical tools to help us understand and describe the universe. Euler also made a monumental contribution to astronomy and orbital mechanics, developing what he called astronomia mechanica. Orbital mechanics of artificial satellites and spacecraft is based on Euler's analysis of astromechanics. However, previous books have often neglected many of his discoveries in this field. For example, orbital mechanics texts refer to the five equilibrium points in the Sun-Earth-Moon system as Lagrange points, failing to credit Euler who first derived the differential equations for the general n-body problem and who discovered the three collinear points in the three-body problem of celestial mechanics. These equilibrium points are essential today in space exploration; the James Webb Space Telescope (successor to the Hubble), for example, now orbits the Sun near L2, one of the collinear points of the Sun-Earth-Moon system, while future missions to study the universe will place observatories in orbit around Sun-Earth and Earth-Moon equilibrium points that should be properly called Euler-Lagrange points. In this book, the author uses Euler's memoirs, correspondence, and other scholarly sources to explore how he established the mathematical groundwork for the rigorous study of motion in our Solar System. The reader will learn how he studied comets and eclipses, derived planetary orbits, and pioneered the study of planetary perturbations, and how, old and blind, Euler put forward the most advanced lunar theory of his time.

Calculus Made Easy is a book on infinitesimal calculus, initially published in 1910 by Silvanus P. Thompson. It is considered a classic and elegant introduction to the subject. The original text continues to be available as of 2023 from Murine Publication. With its epsilon-delta definition, Calculus Made Easy does not use limits. Instead, it uses a way to get close (to any degree of accuracy) to the correct answer based on Leibniz's ideas about infinitesimals. Modern nonstandard analysis and smooth infinitesimal analysis are now formally supporting it.

This book presents the results of experimental and theoretical studies of the destruction of solids under impact, explosion, high pressures, and strain rates. The content identifies the basic laws of the destruction of bodies under dynamic loads. The results of numerical studies were obtained using numerical methods on the Lagrangian, Euler, and ALE approaches to the description of the motion of continuous media. Numerical methods and mathematical models have been tested by comparison with experimental data and well-known analytical solutions (for instance, Rankin-Hugoniot laws). Experimental studies were performed on unique ballistic installations with the registration of fast processes (high-speed shooting). The results are used as new tests to verify the developing modeling methods. The research objects were metal multilayer plates, functionally graded materials, advanced, smart, and natural materials, etc. The book is interesting to specialists in the field of mathematical modeling and experimental methods for studying fast processes under dynamic loading.

This textbook introduces readers to the fundamental notions of modern probability theory. The only prerequisite is a working knowledge in real analysis. Highlighting the connections between martingales and Markov chains on one hand, and Brownian motion and harmonic functions on the other, this book provides an introduction to the rich interplay between probability and other areas of analysis.Arranged into three parts, the book begins with a rigorous treatment of measure theory, with applications to probability in mind. The second part of the book focuses on the basic concepts of probability theory such as random variables, independence, conditional expectation, and the different types of convergence of random variables. In the third part, in which all chapters can be read independently, the reader will encounter three important classes of stochastic processes: discrete-time martingales, countable state-space Markov chains, and Brownian motion. Each chapter ends with a selection of illuminating exercises of varying difficulty. Some basic facts from functional analysis, in particular on Hilbert and Banach spaces, are included in the appendix. Measure Theory, Probability, and Stochastic Processes is an ideal text for readers seeking a thorough understanding of basic probability theory. Students interested in learning more about Brownian motion, and other continuous-time stochastic processes, may continue reading the author's more advanced textbook in the same series (GTM 274).

This textbook provides an introduction to methods for solving nonlinear partial differential equations (NLPDEs). After the introduction of several PDEs drawn from science and engineering, readers are introduced to techniques to obtain exact solutions of NLPDEs. The chapters include the following topics: Nonlinear PDEs are Everywhere; Differential Substitutions; Point and Contact Transformations; First Integrals; and Functional Separability. Readers are guided through these chapters and are provided with several detailed examples. Each chapter ends with a series of exercises illustrating the material presented in each chapter. This Second Edition includes a new method of generating contact transformations and focuses on a solution method (parametric Legendre transformations) to solve a particular class of two nonlinear PDEs.

This contributed volume showcases the most significant results obtained from the DFG Priority Program on Compressed Sensing in Information Processing.  Topics considered revolve around timely aspects of compressed sensing with a special focus on applications, including compressed sensing-like approaches to deep learning; bilinear compressed sensing - efficiency, structure, and robustness; structured compressive sensing via neural network learning; compressed sensing for massive MIMO; and security of future communication and compressive sensing.