Regning og matematisk analyse

Kvalitative analyseprocesser giver konkret indblik i, hvordan du kan analysere empirisk materiale. Bogen starter der, hvor mange metodebøger stopper: Når du har dit empiriske materiale i hus, hvad gør du så? Analysearbejdet er helt centralt for kvalitativ forskning – det er gennem analyserne, at resultaterne fremkommer. Alligevel er det ofte ret dunkelt, hvordan analysen egentlig er foretaget. Denne bog sætter spot på selve analyseprocessen, de analytiske greb, som er anvendt, og hvordan teoretisk begrebsarbejde har guidet analysen undervejs. På den måde inviterer bogen læseren ind i forskerens værksted og retter fokus på selve analyseprocessen med dens kringlede proces, trin og vildveje, tilrettelæggelse og strategier.Bogen er inddelt i hermeneutiske, poststrukturalistiske og dialektiske analyseperspektiver. Det er imidlertid en væsentlig pointe, at det teoretiske afsæt ikke på nogen entydig måde giver én bestemt analysemetode. Derfor præsenterer bogen forskellige analysetilgange under hver af de tre overskrifter og viser, hvordan man på forskellige måder kan arbejde sig igennem den analytiske proces. Forfatterne tilbyder desuden anvisninger i form af modeller, analytikker og spørgsmål, som man kan lade sig inspirere af, når man skal planlægge og gennemføre sine egne analyser.Kvalitative analyseprocesser henvender sig primært til studerende, der skal analysere kvalitativt empirisk materiale som led i en lang eller mellemlang videregående uddannelse eller et ph.d.-forløb. Den er især anvendelig indenfor det pædagogisk-psykologiske felt og beslægtede områder. De beskrevne analyseprocesser har dog samme relevans udover disse felter, og bogen kan derfor anvendes af alle, som ønsker at blive klædt bedre på til at gennemføre teoretisk informerede kvalitative analyser.

Mål- og integralteori har op igennem det 20. århundrede udviklet sig til at udgøre en væsentlig del af grundlaget for matematiske discipliner som f.eks. sandsynlighedsteori, statistik, matematisk fysik, funktionel-analyse og matematisk finansiering. Denne bog giver en stringent indføring i de grundlæggende elementer af mål- og integralteorien, og den kan således danne et solidt udgangspunkt for videregående studier inden for de netop nævnte matematiske discipliner."Grundlæggende mål- og integralteori" henvender sig som udgangspunkt til læsere med en matematisk baggrund svarende til et typisk første studieår på matematik-studiet ved et dansk universitet. Basale resultater og teknikker i forbindelse med grænseovergang og kontinuitet forventes således at være bekendte af læseren, men derudover opbygges teorien gennemgående fra bunden uden yderligere forudsætninger.

Bogen er et redskab til matematiklærere i arbejdet med at støtte elever i matematikvanskeligheder på mellemtrinnet.Når en elev er gået i stå i matematikundervisningen, kan læreren med dette redskab gennemføre en samtale med eleven og kortlægge elevens problemer. Kortlægningen kan inspirere læreren til handlinger, der styrker og udviklereleven positivt i matematikundervisningen. Målet er ogsåat modvirke elevens negative oplevelser og vanskelighederi faget. Materialet har fokus på de ændringer i forventninger, krav og mål, som eleverne møder i skiftet fra indskoling til mellemtrin. Samtidig er der fokus på, hvordan man som lærer opdager, kortlægger og afhjælper regnehuller i forhold til elevens færdigheder og viden.Svage præstationer i matematik kan få alvorlige personlige og sociale konsekvenser. Alle elever har derfor krav på at blive mødt med lærings- og undervisningsbetingelser, der forebygger og afhjælper matematikvanskeligheder. Materialets tilgang bygger på forfatternes begreb ’regnehuller’,der bl.a. indebærer, at matematikvanskeligheder kan imødegås med mange forskellige strategier, og at den bedste strategi ikke altid er at starte med ’at fylde hullet op’.Bogen er en del af serien Matematikvanskeligheder, der dækker grundlæggende matematik på alle alders- og klassetrin. De tre bøger har samme tilgang til undervisning af elever med matematikvanskeligheder. I denne bog er der særligt fokus på samtale i matematikundervisningen og på den direkte dialog mellem lærer og elev, når læreren skal undersøge, hvordan en elev er kommet i vanskeligheder med matematik. Erfaringer viser, at der er en stor spredning af elevers begrebsudvikling i matematik. Mange elever følger ikke den lineære progression, der er indbygget i klassetrin eller alder i den officielle beskrivelse af fagformål. Derfor kan bogen med fordel bruges i arbejdet med elever på andre klassetrin, selvom det fagfaglige udgangspunkt er mellemtrinnet.Bogen henvender sig til matematiklærere og matematikvejledere i grundskolen. Alle, der arbejder med elever i matematikvanskeligheder på ungdomsuddannelser, efterskoler, i støttecentre eller på specialskoler, vil også kunne finde inspiration i bogen.

Træn dine regnefærdigheder her. Der er regnestykker, hvor man skal lægge sammen, dividere, trække fra, gange, udregne procent og inden for logik.Dette øvehæfte, som man kan skrive i, indeholder opgaver af forskellig sværhedsgrad. Det er velegnet fra ca. 5. klasse og op. Hæftet er også meget velegnet til ungdoms- og erhvervsuddannelserne for at træne og holde regnekundskaberne ved lige.Det er sjovt at regne. Hold hjernen frisk! Til regnestykkerne i denne bog kan man enten træne hovedregning, bruge papir og blyant eller regnemaskine.God fornøjelse!

"Differential Equations" by Dr. Upton is a captivating journey through the mathematical landscapes of calculus, offering a rich exploration of historical context, first and second-order equations, systems, Laplace transforms, numerical methods, and real-world applications. With insightful content and advanced topics, this book inspires a deep understanding of complex concepts while providing practical applications in science and engineering.

Dieses essential stellt in kondensierter Form eine Neuinterpretation der Weierstraß'schen Konstruktion der reellen Zahlen vor: Ein vergleichsweise neuer Quellenfund lässt darauf schließen, dass der Weierstraß¿sche Zahlbegriff bereits auf Mengenbegriffen basierte und somit sehr viel elementarer ist, als bislang angenommen wurde.Die beiden bislang bekanntesten Alternativdefinitionen der reellen Zahlen ¿ mittels rationaler Folgen und Konvergenz (Cantor) bzw. als Segmente (Dedekind) ¿ werden hier ebenfalls kurz erläutert und mit der Weierstraß¿schen Konstruktion verglichen.Eine ausführliche Darstellung anhand der Originalquellen findet sich in Spalt, Die Grundlegung der Analysis durch Karl Weierstraß (Springer Spektrum, 2022).

The monograph is devoted to the construction of the high-order finite difference and finite element methods for numerical solving multidimensional boundary-value problems (BVPs) for different partial differential equations, in particular, linear Helmholtz and wave equations, nonlinear Burgers¿ equations, and elliptic (Schrödinger) equation. Despite of a long history especially in development of the theoretical background of these methods there are open questions in their constructive implementation in numerical solving the multidimensional BVPs having additional requirement on physical parameters or desirable properties of its approximate solutions. Over the last two decades many papers on this topics have been published, in which new constructive approaches to numerically solving the multidimensional BVPs were proposed, and its highly desirable to systematically collect these results. This motivate us to write thus monograph based on our research results obtainedin collaboration with the co-authors. Since the topic is importance we believe that this book will be useful to readers, graduate students and researchers interested in the field of computational physics, applied mathematics, numerical analysis and applied sciences

This book aims to establish a systematic theory on the synchronization for wave equations with locally distributed controls. It is structured in two parts. Part I is devoted to internal controls, while Part II treats the case of mixed internal and boundary controls. The authors present necessary mathematical formulations and techniques for analyzing and solving problems in this area. They also give numerous examples and applications to illustrate the concepts and demonstrate their practical relevance. The book provides an overview of the field and offers an in-depth analysis of new results with elegant proofs. By reading this book, it can be found that due to the use of internal controls, more deep-going results on synchronization can be obtained, which makes the corresponding synchronization theory more precise and complete.Graduate students and researchers in control and synchronization for partial differential equations, functional analysis find this book useful. It is also an excellent reference in the field. Thanks to the explicit criteria given in this book for various notions of controllability and synchronization, researchers and practitioners can effectively use the control strategies described in this book and make corresponding decisions regarding system design and operation.

Statistical Analysis for Civil Engineers: Mathematical Theory and Applied Experiment Design is a well-researched and topically organized reference book that guides its readers, both in academia and industry, to recognize how to describe unpredictable events in a quantitative way and to learn how these events can be incorporated into practical engineering analysis that facilitates data-driven problem solving and optimization-based decision-making.Written by experts in the field with a proven track record as educators and practicing consultancy specialists, this book has been developed in such a manner that it advances understanding of the mathematical theory underlying analytical methodology gradually. It also supports practical application through relevant worked examples in a variety of civil engineering branches, notably structural, materials, transportation, and geotechnical engineering. Through all stages of data analysis, numerical modeling and simulation, and implementation, the volume emphasizes the need to change the current perception with respect to the use of modern statistical techniques in the scientific as well as practical spheres of civil engineering.

This official Student Solutions Manual includes solutions to the odd-numbered exercises featured in the third edition of Steven Strogatz's classic text Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering.

This book aims at introducing students into the modern analytical foundations to treat problems and situations in the Calculus of Variations solidly and rigorously. Since no background is taken for granted or assumed, as the textbook pretends to be self-contained, areas like basic Functional Analysis and Sobolev spaces are studied to the point that chapters devoted to these topics can be utilized by themselves as an introduction to these important parts of Analysis. The material in this regard has been selected to serve the needs of classical variational problems, leaving broader treatments for more advanced and specialized courses in those areas. It should not be forgotten that problems in the Calculus of Variations historically played a crucial role in pushing Functional Analysis as a discipline on its own right. The style is intentionally didactic. After a first general chapter to place optimization problems in infinite-dimensional spaces in perspective, the first part of the book focuses on the initial important concepts in Functional Analysis and introduces Sobolev spaces in dimension one as a preliminary, simpler case (much in the same way as in the successful book of H. Brezis). Once the analytical framework is covered, one-dimensional variational problems are examined in detail including numerous examples and exercises. The second part dwells, again as a first-round, on another important chapter of Functional Analysis that students should be exposed to, and that eventually will find some applications in subsequent chapters. The first chapter of this part examines continuous operators and the important principles associated with mappings between functional spaces; and another one focuses on compact operators and their fundamental and remarkable properties for Analysis. Finally, the third part advances to multi-dimensional Sobolev spaces and the corresponding problems in the Calculus of Variations. In this setting, problems become much more involved and, for this same reason, much more interesting and appealing. In particular, the final chapter dives into a number of advanced topics, some of which reflect a personal taste. Other possibilities stressing other kinds of problems are possible. In summary, the text pretends to help students with their first exposure to the modern calculus of variations and the analytical foundation associated with it. In particular, it covers an extended introduction to basic functional analysis and to Sobolev spaces. The tone of the text and the set of proposed exercises will facilitate progressive understanding until the need for further challenges beyond the topics addressed here will push students to more advanced horizons.

The book concerns with solving about 650 ordinary and partial differential equations. Each equation has at least one solution and each solution has at least one coloured graph. The coloured graphs reveal different features of the solutions. Some graphs are dynamical as for Clairaut differential equations. Thus, one can study the general and the singular solutions. All the equations are solved by Mathematica. The first chapter contains mathematical notions and results that are used later through the book. Thus, the book is self-contained that is an advantage for the reader. The ordinary differential equations are treated in Chapters 2 to 4, while the partial differential equations are discussed in Chapters 5 to 10. The book is useful for undergraduate and graduate students, for researchers in engineering, physics, chemistry, and others. Chapter 9 treats parabolic partial differential equations while Chapter 10 treats third and higher order nonlinear partial differential equations, both with modern methods. Chapter 10 discusses the Korteweg-de Vries, Dodd-Bullough-Mikhailov, Tzitzeica-Dodd-Bullough, Benjamin, Kadomtsev-Petviashvili, Sawada-Kotera, and Kaup-Kupershmidt equations.

This book contains an Irish translation of the the second-edition English version (ISBN 9780244879822, https://logicpress.ie/2020-1/index.html) of a one-semester undergraduate course taught through the medium of Irish. I have given this course in Maynooth each year since 2006. The class is typically quite mixed, and the book aims to challenge students familiar with rigorous arguments while simultaneously providing comfort to other students that prefer procedural mathematics---not an easy balancing act! For instance, some exercises are targeted at one or other of these two groups of students and, while some material is done in the context of metric spaces, guidance is given to the reader who prefers to stick to the real line. This second edition of the book has no major new topics, but it is about twice as long as the first. Some topics are explored in greater depth (e.g. conjugacy and semiconjugacy, bifurcation) and some proofs have been added (notably for Sharkovsky's and Singer's theorems). Additional exercises have been given, and the most challenging ones are marked appropriately. There is a glossary giving translations of terms.

Dieses Buch beschreibt aus rein epidemiologischer Sicht, die Entstehung und Entwicklung einer Epidemie mit Hilfe von Differentialgleichungssystemen. Dabei wird die Bevölkerung in die bekannten Kompartimente oder Klassen aufgeteilt. Ausgehend vom einfachsten Modell werden in nachvollziehbaren kleinen Schritten die bestehenden Modelle erweitert, um Phänomene wie Rückfall oder Immunitätsverlust zu modellieren. Zudem werden in weiteren Schritten Kompartimente hinzugefügt, die mit der Berücksichtigung von Quarantäne und Impfung einhergehen. Jedes Modell wird vollständig analysiert und die Ergebnisse festgehalten. Danach folgt für jedes Modell mindestens ein vollständig gelöstes Zahlenbeispiel inklusive einer Darstellung für den jeweiligen Epidemieverlauf. Kern dieses Buches bilden die Simulationen und Prognosen für vier verschiedene Covid-Pandemiewellen in Zentraleuropa der letzten Jahre mit den erfassten Daten und unter Verwendung von 6 Modellen. Darüber hinaus werden Möglichkeiten zur Schätzung von Raten und Anfangswerten präsentiert, die für eine Vorhersage eines Epidemieverlaufs unerlässlich sind. Dieses Buch ist wegweisend für den Einstieg in die Modellierung von Pandemien und eignet sich auch als Nachschlagewerk.

This book gathers the latest advances, innovations, and applications in the field of computational engineering, as presented by leading international researchers and engineers at the 29th International Conference on Computational & Experimental Engineering and Sciences (ICCES), held in Shenzhen, China on May 26-29, 2023. ICCES covers all aspects of applied sciences and engineering: theoretical, analytical, computational, and experimental studies and solutions of problems in the physical, chemical, biological, mechanical, electrical, and mathematical sciences. As such, the book discusses highly diverse topics, including composites; bioengineering & biomechanics; geotechnical engineering; offshore & arctic engineering; multi-scale & multi-physics fluid engineering; structural integrity & longevity; materials design & simulation; and computer modeling methods in engineering. The contributions, which were selected by means of a rigorous international peer-review process, highlight numerous exciting ideas that will spur novel research directions and foster multidisciplinary collaborations.

These are the proceedings of the 27th International Conference on Domain Decomposition Methods in Science and Engineering, which was held in Prague, Czech Republic, in July 2022.Domain decomposition methods are iterative methods for solving the often very large systems of equations that arise when engineering problems are discretized, frequently using finite elements or other modern techniques. These methods are specifically designed to make effective use of massively parallel, high-performance computing systems.The book presents both theoretical and computational advances in this domain, reflecting the state of art in 2022.

This textbook is an introduction to the methods needed to solve partial differential equations (PDEs). Readers are introduced to PDEs that come from a variety of fields in engineering and the natural sciences. The chapters include the following topics: First Order PDEs, Second Order PDEs, Fourier Series, Separation of Variables, the Fourier Transform, and higher dimensional problems. Readers are guided through these chapters where techniques for solving first and second order PDEs are introduced. Each chapter ends with series of exercises to facilitate learning as well as illustrate the material presented in each chapter.

This book provides an alternative approach to time-independent perturbation theory in non-relativistic quantum mechanics. It allows easy application to any initial condition because it is based on an approximation to the evolution operator and may also be used on unitary evolution operators for the unperturbed Hamiltonian in the case where the eigenvalues cannot be found. This flexibility sets it apart from conventional perturbation theory. The matrix perturbation method also gives new theoretical insights; for example, it provides corrections to the energy and wave function in one operation. Another notable highlight is the facility to readily derive a general expression for the normalization constant at m-th order, a significant difference between the approach within and those already in the literature. Another unique aspect of the matrix perturbation method is that it can be extended directly to the Lindblad master equation. The first and second-order corrections are obtained for this equation and the method is generalized for higher orders. An alternative form of the Dyson series, in matrix form instead of integral form, is also obtained. Throughout the book, several benchmark examples and practical applications underscore the potential, accuracy and good performance of this novel approach. Moreover, the method's applicability extends to some specific time-dependent Hamiltonians. This book represents a valuable addition to the literature on perturbation theory in quantum mechanics and is accessible to students and researchers alike.

Kolmogorov equations are a fundamental bridge between the theory of partial differential equations and that of stochastic differential equations that arise in several research fields.This volume collects a selection of the talks given at the Cortona meeting by experts in both fields, who presented the most recent developments of the theory. Particular emphasis has been given to degenerate partial differential equations, Itô processes, applications to kinetic theory and to finance.

This book helps the reader make use of the mathematical models of biological phenomena starting from the basics of programming and computer simulation. Computer simulations based on a mathematical model enable us to find a novel biological mechanism and predict an unknown biological phenomenon. Mathematical biology could further expand the progress of modern life sciences. Although many biologists are interested in mathematical biology, they do not have experience in mathematics and computer science. An educational course that combines biology, mathematics, and computer science is very rare to date. Published books for mathematical biology usually explain the theories of established mathematical models, but they do not provide a practical explanation for how to solve the differential equations included in the models, or to establish such a model that fits with a phenomenon of interest. MATLAB is an ideal programming platform for the beginners of computer science. This book starts from the very basics about how to write a programming code for MATLAB (or Octave), explains how to solve ordinary and partial differential equations, and how to apply mathematical models to various biological phenomena such as diabetes, infectious diseases, and heartbeats. Some of them are original models, newly developed for this book. Because MATLAB codes are embedded and explained throughout the book, it will be easy to catch up with the text. In the final chapter, the book focuses on the mathematical model of the proneural wave, a phenomenon that guarantees the sequential differentiation of neurons in the brain. This model was published as a paper from the author¿s lab (Sato et al., PNAS 113, E5153, 2016), and was intensively explained in the book chapter ¿Notch Signaling in Embryology and Cancer¿, published by Springer in 2020. This book provides the reader who has a biological background with invaluable opportunities to learn and practice mathematical biology.

This book focuses on Krylov subspace methods for solving linear systems, which are known as one of the top 10 algorithms in the twentieth century, such as Fast Fourier Transform and Quick Sort (SIAM News, 2000). Theoretical aspects of Krylov subspace methods developed in the twentieth century are explained and derived in a concise and unified way. Furthermore, some Krylov subspace methods in the twenty-first century are described in detail, such as the COCR method for complex symmetric linear systems, the BiCR method, and the IDR(s) method for non-Hermitian linear systems.The strength of the book is not only in describing principles of Krylov subspace methods but in providing a variety of applications: shifted linear systems and matrix functions from the theoretical point of view, as well as partial differential equations, computational physics, computational particle physics, optimizations, and machine learning from a practical point of view.The book is self-contained in that basic necessary concepts of numerical linear algebra are explained, making it suitable for senior undergraduates, postgraduates, and researchers in mathematics, engineering, and computational science. Readers will find it a useful resource for understanding the principles and properties of Krylov subspace methods and correctly using those methods for solving problems in the future.