Matematik

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.

The Instant New York Times Bestseller“Engaging and entertaining… a glimpse of the economy of the future.” —Tim Wu, New York Times Book ReviewFrom the New York Times bestselling author of The Signal and the Noise, the definitive guide to our era of risk—and the players raising the stakes   In the bestselling The Signal and the Noise, Nate Silver showed how forecasting would define the age of Big Data. Now, in this timely and riveting new book, Silver investigates “the River,” the community of like-minded people whose mastery of risk allows them to shape—and dominate—so much of modern life. These professional risk-takers—poker players and hedge fund managers, crypto true believers and blue-chip art collectors—can teach us much about navigating the uncertainty of the twenty-first century. By immersing himself in the worlds of Doyle Brunson, Peter Thiel, Sam Bankman-Fried, Sam Altman, and many others, Silver offers insight into a range of issues that affect us all, from the frontiers of finance to the future of AI. Most of us don’t have traits commonly found in the River: high tolerance for risk, appreciation of uncertainty, affinity for numbers—paired with an instinctive distrust of conventional wisdom and a competitive drive so intense it can border on irrational. For those in the River, complexity is baked in, and the work is how to navigate it. People in the River have increasing amounts of wealth and power in our society, and understanding their mindset—and the flaws in their thinking— is key to understanding what drives technology and the global economy today. Taking us behind the scenes from casinos to venture capital firms, and from the FTX inner sanctum to meetings of the effective altruism movement, On the Edge is a deeply reported, all-access journey into a hidden world of power bro­kers and risk-takers.

Petri nets model concurrent and distributed systems where active components communicate through the production and absorption of various kinds of resources. Although the dynamic properties of such systems may be very complex, they may sometimes be connected to the static structure of a Petri net. Many properties are decidable, but their complexity may be huge. It is often opportune to restrict oneself to classes of systems, to partial algorithms, and to similar but simpler properties. Instead of analysing a given system, it is also possible to search for a system satisfying some desired properties by construction. This comprehensive textbook/reference presents and discusses these issues in-depth in the context of one of the most fundamental Petri net models, called place/transition nets. The presentation is fortified by means of many examples and worked exercises. Among topics addressed: ¿ In which order may actions may be generated and scheduled? ¿ What states and configurations may be reached in a concurrent system? ¿ Which interesting classes of systems can be analysed relatively efficiently? ¿ Is it possible to synthesise a system of some class from its behaviour? ¿ How can systems be represented algebraically, compositionally, and concisely? This unique text, based on introductory as well as on advanced courses on distributed systems, will serve as an invaluable guide for students and (future) researchers interested in theoretical¿as well as in practical¿aspects of Petri nets and related system models. Eike Best has been a full professor (now retired) affiliated to Carl von Ossietzky Universität Oldenburg, Germany. Raymond Devillers has been a full professor (now retired) affiliated to Université Libre de Bruxelles, Belgium. The authors have a long record as collaborators in the fields of Petri nets and the semantics of concurrency.

This textbook introduces quantitative methods in operations management, based on operational research. Written for undergraduate and graduate students as well as practitioners, this book serves as a valuable compendium of essential tools for project planning, control, and strategic decision-making.Drawing from the expertise of both experienced scientists and seasoned practical managers, the descriptions of each tool are a harmonious blend of theoretical insights and real-world applicability. With a focus on accessibility, the authors have thoughtfully combined abstract concepts with easy-to-follow examples and detailed case studies.Readers will benefit from the abundance of well-explained recommendations and practical problem-solving approaches, where the book offers guidance on how to solve presented issues by using commercial software. Whether one seeks to refine project management, optimize operations, or make strategic choices, this book equips readers with theknowledge and proficiency required to excel in the dynamic field of operations management.

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 is intended as an introduction to Probability Theory and Mathematical Statistics for students in mathematics, the physical sciences, engineering, and related fields. It is based on the author's 25 years of experience teaching probability and is squarely aimed at helping students overcome common difficulties in learning the subject. The focus of the book is an explanation of the theory, mainly by the use of many examples. Whenever possible, proofs of stated results are provided. All sections conclude with a short list of problems. The book also includes several optional sections on more advanced topics. This textbook would be ideal for use in a first course in Probability Theory. Contents: Probabilities Conditional Probabilities and Independence Random Variables and Their Distribution Operations on Random Variables Expected Value, Variance, and Covariance Normally Distributed Random Vectors Limit Theorems Introduction to Stochastic Processes Mathematical Statistics Appendix Bibliography Index

From zero to infinity, this entertaining hardback guide opens up a new world of knowledge based on the magic of numbers. Numbers have occupied our thoughts since man first realized he had not one opposable thumb but two. And from simple enumeration they have grown to be the most important and universal language there is. The Book of Numbers highlights the dominant role that numbers play in everyday life, as well as exploring how numbers and number systems evolved, and delving into the mysteries of mankind's most powerful numbers: - What are the top-ten One Hit Wonders?- What's so magnificent about 7?- Why is 13 unlucky?>From algebra to astrology, music to mythology, from religion to recreation and from science to superstition, The Book of Numbers embraces this infinitely broad subject and puts it all in order-beginning with 0.

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.

This volume contains 8 papers that have been collected by the Canadian Society for History and Philosophy of Mathematics. It showcases rigorously reviewed contemporary scholarship on an interesting variety of topics in the history and philosophy of mathematics.Some of the topics explored include:A way to rethink how logic is taught to philosophy students by using a rejuvenated version of the Aristotelian idea of an argument schemaA quantitative approach using data from Wikipedia to study collaboration between nineteenth-century British mathematiciansThe depiction and perception of Émilie Du Châtelet¿s scientific contributions as viewed through the frontispieces designed for books written by or connected to herA study of the Cambridge Women¿s Research Club, a place where British women were able to participate in scholarly scientific discourse in the middle of the twentieth centuryAn examination of the researchand writing process of mathematicians by looking at their drafts and other preparatory notesA global history of al-Khw¿r¿zm¿¿s Kit¿b al-jabr wa-l-muq¿bala as obtained by tracing its reception through numerous translations and commentariesWritten by leading scholars in the field, these papers are accessible not only to mathematicians and students of the history and philosophy of mathematics, but also to anyone with a general interest in mathematics.

This book is an introduction to machine learning, with a strong focus on the mathematics behind the standard algorithms and techniques in the field, aimed at senior undergraduates and early graduate students of Mathematics. There is a focus on well-known supervised machine learning algorithms, detailing the existing theory to provide some theoretical guarantees, featuring intuitive proofs and exposition of the material in a concise and precise manner. A broad set of topics is covered, giving an overview of the field. A summary of the topics covered is: statistical learning theory, approximation theory, linear models, kernel methods, Gaussian processes, deep neural networks, ensemble methods and unsupervised learning techniques, such as clustering and dimensionality reduction. This book is suited for students who are interested in entering the field, by preparing them to master the standard tools in Machine Learning. The reader will be equipped to understand the main theoretical questions of the current research and to engage with the field.

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.

"This collection of four short courses looks at group representations, graph spectra, statistical optimality, and symbolic dynamics, highlighting their common roots in linear algebra. Aimed at researchers and beginning Ph.D. students, it includes copious exercises, notes, and references, leading the reader from the basics to high-level applications"--

From the bestselling author of Quantum Computing for Everyone, a concise, accessible, and elegant approach to mathematics that not only illustrates concepts but also conveys the surprising nature of the digital information age.Most of us know something about the grand theories of physics that transformed our views of the universe at the start of the twentieth century: quantum mechanics and general relativity. But we are much less familiar with the brilliant theories that make up the backbone of the digital revolution. In Beautiful Math, Chris Bernhardt explores the mathematics at the very heart of the information age. He asks questions such as: What is information? What advantages does digital information have over analog? How do we convert analog signals into digital ones? What is an algorithm? What is a universal computer? And how can a machine learn?The four major themes of Beautiful Math are information, communication, computation, and learning. Bernhardt typically starts with a simple mathematical model of an important concept, then reveals a deep underlying structure connecting concepts from what, at first, appear to be unrelated areas. His goal is to present the concepts using the least amount of mathematics, but nothing is oversimplified. Along the way, Bernhardt also discusses alphabets, the telegraph, and the analog revolution; information theory; redundancy and compression; errors and noise; encryption; how analog information is converted into digital information; algorithms; and finally, neural networks. Historical anecdotes are included to give a sense of the technology at that time, its impact, and the problems that needed to be solved. Taking its readers by the hand, regardless of their math background, Beautiful Math is a fascinating journey through the mathematical ideas that undergird our everyday digital interactions.

Mathematik für die Informatik: Dieses Buch bringt Ihnen die Grundlagen beiDieses Lehrbuch bietet eine Einführung in die grundlegenden Begriffe und Strukturen der Mathematik, welche am Anfang eines Informatikstudiums relevant sind. Weiterhin demonstriert es Anwendungen von mathematischen Konzepten und Methoden in der Informatik. Diese betreffen insbesondere formale Methoden der Programmverifikation und -entwicklung und den Entwurf von generischen Programmen. Ein spezielles Konzept mit einer leicht verständlichen Vermittlung des Stoffs, vielen Beispielen mit Rückgriffen auf die Schulmathematik und detaillierten Beweisen (verbunden mit der Erklärung des logischen Hintergrunds) erleichtert den Einstieg in die Mathematik an einer wissenschaftlichen Hochschule. Dadurch werden die Studierenden auch auf spätere Begriffe und tiefergehende Anwendungen der Mathematik in der Informatik gut vorbereitet. Die 5. Auflage erweitert den Inhalt um die Grundlagen der linearen Algebra. Die Übungsaufgaben zu den einzelnen Kapitel helfen, das Erlernte zu festigen und zu kontrollieren. Zahlreiche Lösungsvorschläge am Ende des Buchs ermöglichen die Überprüfung der eigenen Lösungen.Der InhaltMengentheoretische Grundlagen ¿ Logische Grundlagen ¿ Allgemeine direkte Produkte und Datenstrukturen ¿ Mathematische Beweise ¿ Spezifikation und Programmverifikation ¿ Spezielle Funktionen ¿ Spezielle Relationen und gerichtete Graphen ¿ Elementare Kombinatorik und ungerichtete Graphen ¿ Diskrete Wahrscheinlichkeitstheorie ¿ Generische Programmierung ¿ Grundbegriffe algebraischer Strukturen ¿ Formale Einführung der natürlichen Zahlen - Grundbegriffe der linearen AlgebraMit diesem Buch gelingt der Einstieg ins Informatik-StudiumMit diesem Buch schaffen Sie eine solide Basis für die Mathematikausbildung im Rahmen desInformatikstudiums. Zudem sind Sie durch die vorgestellten Problemstellungen in der Lage, selbstständig mathematische Konzepte und Methoden anzuwenden. Zielgruppen dieses Buchs über Mathematik in der Informatik sind Bachelor-Studierende in den ersten Studiensemestern folgender Fachbereiche:InformatikMathematikIngenieurwissenschaften

Unlock the secrets of the universe with Schrödinger Equation - Fundamental Aspects and Potential Applications. Delve into the heart of quantum mechanics, where matter, energy, and mathematics intertwine in a dance of profound discovery. This essential volume introduces you to the spectral theory of the Schrödinger equation, offering a sturdy foundation to explore its enigmatic depths. Discover the fascinating world of scattering theory, unraveling the intricacies of quantum interactions, while the principles of quantization and Feynman path integrals reveal the mechanics of quantum systems. With a fresh perspective, we explore relative entropy methods and transformation theory, unveiling their significance in crafting singular diffusion processes akin to Schrödinger equations. This well-organized and accessible book caters to a diverse audience, from students and researchers to professionals in functional analysis, probability theory, and quantum dynamics. Within these pages, yoüll uncover the profound wonders of the Schrödinger equation and its vast potential in science, engineering, and technology. Embark on a journey through the quantum cosmos and let your understanding of the universe expand as you explore the quantum realm. Welcome to a world where matter and energy dance to the tune of Schrödinger¿s equation, a world filled with infinite possibilities and extraordinary insights.

This workbook is designed for elementary students, specifically those in 3rd through 5th grade, and focuses on multiplication tables from 1 to 12. Packed with interactive exercises and engaging games, your child is sure to enjoy learning. Additional practice sheets are included for those who may need extra reinforcement or wish to involve multiple family members. If you require more copies, feel free to contact us, and for a small fee, additional copies can be provided!Key Features of "The Math Guru, Multiplication Edition Volume 1 Workbook:Interactive Learning Experience: The workbook offers an interactive approach to multiplication, turning what could be a daunting subject into an exciting adventure. With colorful visuals, engaging activities, and lively characters, students are encouraged to actively participate in their learning journey.Diverse Exercises: From basic multiplication tables, the workbook covers a wide range of exercises that cater to different learning levels. This diversity ensures that students can progress at their own pace, building a strong foundation in multiplication.Exciting Games: Learning becomes play with the inclusion of interactive games that reinforce multiplication skills in a dynamic and entertaining manner. These games not only enhance mathematical proficiency but also make the workbook an enjoyable companion for young learners.

Il presente libro trae origine dalle lezioni del corso di Geometria e vuole essere un utile strumento per la preparazione agli esami presenti in diversi corsi di laurea triennale, quali, Architettura e Ingegneria. Gli esercizi scelti, prima di tutto, suggeriscono percorsi per approfondimenti e riflessioni, personali, sulle nozioni teoriche da studiare per gli esami. Inoltre, sono stati elaborati in maniera tale da indurre il lettore a moderare l'uso dei procedimenti in serie, ripetitivi, applicati in maniera acritica, offrendo strategie per trovare soluzioni più dirette ed soprattutto ad affinare la capacità di pensiero e ragionamento.