Matematisk modellering

Reinforcement learning is a learning paradigm concerned with learning to control a system so as to maximize a numerical performance measure that expresses a long-term objective. What distinguishes reinforcement learning from supervised learning is that only partial feedback is given to the learner about the learner's predictions. Further, the predictions may have long term effects through influencing the future state of the controlled system. Thus, time plays a special role. The goal in reinforcement learning is to develop efficient learning algorithms, as well as to understand the algorithms' merits and limitations. Reinforcement learning is of great interest because of the large number of practical applications that it can be used to address, ranging from problems in artificial intelligence to operations research or control engineering. In this book, we focus on those algorithms of reinforcement learning that build on the powerful theory of dynamic programming. We give a fairly comprehensive catalog of learning problems, describe the core ideas, note a large number of state of the art algorithms, followed by the discussion of their theoretical properties and limitations. Table of Contents: Markov Decision Processes / Value Prediction Problems / Control / For Further Exploration

This book investigates in detail the mathematical methods and computation methods in efficient solution of some open nonlinear seepage flow problems involved in engineering problems. Developed engineering technologies and some relevant practical field applications are also provided. The introduced open nonlinear problems include nonlinear quadratic pressure gradient term problem, compressible gas seepage flow problem and low-velocity non-Darcy seepage flow problem. Studies on these nonlinear seepage flow problems have attracted engineers and scientists from various disciplines, such as geo-energy engineering, civil and environmental engineering, fluid mechanics, applied mathematics and computation. In particular, the book systematically establishes a fundamental theory for a strongly nonlinear problem of low-velocity non-Darcy seepage flow from a new perspective of moving boundary, while emphasizing the usage of mathematical linearization transformation methods and computational methods into the analytical and numerical solution of the strongly nonlinear partial differential equations. Sufficient knowledge of mathematics is always introduced ahead of model solution to assist readers. And the procedure of strict formula deduction in the model solution process is provided in detail. High-solution figures and tables from model solution are rich in the book. Therefore, it is very helpful for the readers to master the nonlinear model solution methods and engineering technologies. The book is intended for upper undergraduate students and graduate students who are interested in engineering technology, fluid mechanics and applied mathematics, researchers and engineers working on geo-energy science and engineering and field applications.

This book comprises select peer-reviewed articles submitted for the proceedings of the International Conference on Mathematics and Computing (ICMC 2022), held by the School of Advanced Sciences, Vellore Institute of Technology, Vellore, India, in association with Ramanujan Mathematical Society, India, Cryptology Research Society of India and Society for Electronic Transactions and Security, India, from 6-8 January 2022. With an aim to identify the existing challenges in the areas of mathematics and computing, the book emphasizes the importance of establishing new methods and algorithms to address these challenges. The book includes topics on diverse applications of cryptology, network security, cyber security, block chain, IoT, mobile network, data analytics, applied algebra, mathematical analysis, mathematical modelling, fluid dynamics, fractional calculus, multi-optimization, integral equations, dynamical systems, numerical analysis and scientific computing. Divided into five major parts-applied algebra and analysis, fractional calculus and integral equations, mathematical modelling and fluid dynamics, numerical analysis, and computer science and applications-the book is a useful resource for students, researchers and faculty as well as practitioners.

The book focuses on original approaches intended to support the development of biologically inspired cognitive architectures. It bridges together different disciplines, including artificial intelligence, linguistics, neuro- and social sciences, psychology and philosophy of mind, among others. The chapters are based on contributions presented at the 2023 Annual International Conference on Brain-Inspired Cognitive Architectures for Artificial Intelligence (the 14th Annual Meeting of the BICA Society, BICA*AI 2023), organized in collaboration with the University of Ningbo and held on October 13-15, 2023, in Ningbo, China. The book discusses emerging methods, theories and ideas towards the realization of general-purpose humanlike artificial intelligence or fostering a better understanding of the ways the human mind works. It provides engineers, mathematicians, psychologists, computer scientists and other experts with a timely snapshot of recent research and a source of inspiration for future developments in the broadly intended areas of artificial intelligence and biological inspiration.

This book sheds light on systems that learn extensively, with purpose and naturally interact with humans. Improving operations and increasing competitive differentiation among manufacturing organizations by harnessing the power of cognitive abilities, IoT can help build and influence the flow of information¿making the shop floor more cognitive through effective processing, analysis, and operational optimization. Now we are seeing the first-hand potential of cognitive computing¿its ability to transform businesses, governments, and society. The real potential of the cognitive age can be realized by combining data analysis and statistical reasoning of machines with uniquely human qualities, such as self-directed goals, common sense, and moral values, improving operations and increasing competitive differentiation among manufacturing organizations. By harnessing the power of cognitive abilities, IoT can help build and influence the flow of information¿making the shop floor more cognitive through effective processing, analysis, and operational optimization. Cognitive initiatives come in all shapes and sizes, from change to strategy and everything in between. What most successful projects have in common, no matter how ambitious, is they start with a clear view of what technology can do. Therefore, the first job of a cognitive scientists is to gain a firm understanding of cognitive abilities, as presented in this book.

This book starts with the rationale for creating an FCM by contrast to other techniques for participatory modeling, as this rationale is a key element to justify the adoption of techniques in a research paper. Fuzzy cognitive mapping is an active research field with over 20,000 publications devoted to externalizing the qualitative perspectives or ¿mental models¿ of individuals and groups. Since the emergence of fuzzy cognitive maps (FCMs) back in the 80s, new algorithms have been developed to reduce bias, facilitate the externalization process, or efficiently utilize quantitative data via machine learning. It covers the development of an FCM with participants through a traditional in-person setting, drawing from the experience of practitioners and highlighting solutions to commonly encountered challenges. The book continues with introducing principles of simulations with FCMs as a tool to perform what-if scenario analysis, while extending those principles to more elaborated simulation scenarios where FCMs and agent-based modeling are combined. Once an FCM model is obtained, the book then details the analytical tools available for practitioners (e.g., to identify the most important factors) and provides examples to aid in the interpretation of results. The discussion concerning relevant extensions is equally pertinent, which are devoted to increasing the expressiveness of the FCM formalism in problems involving uncertainty. The last four chapters focus on building FCM models from historical data. These models are typically needed when facing multi-output prediction or pattern classification problems. In that regard, the book smoothly guides the reader from simple approaches to more elaborated algorithms, symbolizing the noticeable progress of this field in the last 35 years. Problems, recent references, and functional codes are included in each chapter to provide practice and support further learning from practitioners and researchers.

This thesis deals with storage systems that are organized in last-in-first-out stacks. This approach for storing items is common, for example, in container terminals, and is therefore of great practical interest as well as high relevance in the literature. Furthermore, optimization problems that may arise in this context are presented and addressed by developing novel mathematical model formulations and customized solution methods.

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.

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.

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 book publishes select papers presented at the 4th International Conference on Frontiers in Industrial and Applied Mathematics (FIAM-2021), held at the Sant Longowal Institute of Engineering and Technology, Longowal, Punjab, India, from 21¿22 December 2021. Most of the papers deal with mathematical theory embedded with its applications to engineering and sciences. This book illustrates numerical simulation of scientific problems and the state-of-the-art research in industrial and applied mathematics, including various computational and modeling techniques with case studies and concrete examples. Graduate students and researchers, who are interested in real applications of mathematics in the areas of computational and theoretical fluid dynamics, solid mechanics, optimization and operations research, numerical analysis, bio-mathematics, fuzzy, control and systems theory, dynamical systems and nonlinear analysis, algebra and approximation theory, will find the book useful.

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 book presents a new way of thinking about quantum mechanics and machine learning by merging the two. Quantum mechanics and machine learning may seem theoretically disparate, but their link becomes clear through the density matrix operator which can be readily approximated by neural network models, permitting a formulation of quantum physics in which physical observables can be computed via neural networks. As well as demonstrating the natural affinity of quantum physics and machine learning, this viewpoint opens rich possibilities in terms of computation, efficient hardware, and scalability. One can also obtain trainable models to optimize applications and fine-tune theories, such as approximation of the ground state in many body systems, and boosting quantum circuits¿ performance. The book begins with the introduction of programming tools and basic concepts of machine learning, with necessary background material from quantum mechanics and quantum information also provided. This enables the basic building blocks, neural network models for vacuum states, to be introduced. The highlights that follow include: non-classical state representations, with squeezers and beam splitters used to implement the primary layers for quantum computing; boson sampling with neural network models; an overview of available quantum computing platforms, their models, and their programming; and neural network models as a variational ansatz for many-body Hamiltonian ground states with applications to Ising machines and solitons. The book emphasizes coding, with many open source examples in Python and TensorFlow, while MATLAB and Mathematica routines clarify and validate proofs. This book is essential reading for graduate students and researchers who want to develop both the requisite physics and coding knowledge to understand the rich interplay of quantum mechanics and machine learning.

"Packed with examples and illustrations to describe complex concepts and mathematical derivations, and including end-of-chapter problems with helpful hints, this thorough and accessible textbook is the ideal introduction to continuum thermodynamics for senior undergraduate and graduate students of mechanical, aeronautical and civil engineering"--

This book keeps an eye in the direction of applications of advanced and high performance scientific computing in describing the behavior of natural and constructed systems, e.g. chaos, bifurcation, fractal, Lyapunov exponent, period doubling, Poincaré map, strange attractor etc. With the aid of powerful computers the modem theory of chaos and its geometry, the fractals, and attractors are developed. The concepts of object oriented computing are introduced early in the text and steadily expanded as one progresses through the chapters. The beginning of each chapter is of an introductory nature, followed by practical applications, the discussion of numerical results, theoretical investigations on nonlinear stability and convergence.This is the first complete introduction to process modelling and computing that fully integrates software tools ¿ enabling professionals and students to master critical techniques hands on through computer simulations based on the popular MATLAB environment. The book offers a simple tool for all those oscillations that are travelling through the world, helping them discover its hidden beauty.Many applications as well as results of computer simulations are presented. The center of concern is set on existing as well as emerging continuous methods of investigations useful for researchers, engineers and practitioners active in many and often interdisciplinary fields, where physics, electrochemistry, biology and medicine play a key role. Coverage includes:¿ Dynamic behavior of nonlinear systems,¿ Fundamental descriptions of processes exhibiting nonlinear oscillations,¿ Mechanism and function of structures of nonlinear oscillations¿ patterns,¿ Analysis of dynamical oscillations in electric circuits and systems,¿ Artificial intelligence models of natural systems,¿ Nonlinear oscillations in chemistry, biologyand medicine,¿ Oscillations in mechanics and transport systems,¿ Oscillations in fractional-order systems,¿ Energy harvesting systems from the surrounding environment.With an insatiable appetite for exploring the surrounding world and doing research, this book can help readers quickly find ways to use new computers and facilitate the quest for greater knowledge and understanding of reality. The reach of novelty of the book ranges from new mathematical ideas to motivating questions and science issues in many subject areas.

This contributed volume explores innovative research in the modeling, simulation, and control of crowd dynamics. Chapter authors approach the topic from the perspectives of mathematics, physics, engineering, and psychology, providing a comprehensive overview of the work carried out in this challenging interdisciplinary research field. The volume begins with an overview of analytical problems related to crowd modeling. Attention is then given to the importance of considering the social and psychological factors that influence crowd behavior ¿ such as emotions, communication, and decision-making processes ¿ in order to create reliable models. Finally, specific features of crowd behavior are explored, including single-file traffic, passenger movement, modeling multiple groups in crowds, and the interplay between crowd dynamics and the spread of disease.Crowd Dynamics, Volume 4 is ideal for mathematicians, engineers, physicists, and other researchers working in the rapidly growing field of modeling and simulation of human crowds.

This book offers a careful selection of studies in optimization techniques based on artificial intelligence, applied to inverse problems in radiative transfer. In this book, the reader will find an in-depth exploration of heuristic optimization methods, each meticulously described and accompanied by historical context and natural process analogies.From simulated annealing and genetic algorithms to artificial neural networks, ant colony optimization, and particle swarms, this volume presents a wide range of heuristic methods. Additional approaches such as generalized extreme optimization, particle collision, differential evolution, Luus-Jaakola, and firefly algorithms are also discussed, providing a rich repertoire of tools for tackling challenging problems.While the applications showcased primarily focus on radiative transfer, their potential extends to various domains, particularly nonlinear and large-scale problems where traditional deterministic methods fall short. With clear and comprehensive presentations, this book empowers readers to adapt each method to their specific needs. Furthermore, practical examples of classical optimization problems and application suggestions are included to enhance your understanding.This book is suitable to any researcher or practitioner whose interests lie on optimization techniques based in artificial intelligence and bio-inspired algorithms, in fields like Applied Mathematics, Engineering, Computing, and cross-disciplinary areas.

This book describes a new strategy for rehabilitation from injury and/or disease using Crucial Event Therapy. Recent studies have shown that individuals can recuperate more rapidly from surgery and other invasive procedures intended to correct the negative effects of disease or injury through the use of life support systems that operate at the body's natural biofrequencies. The same observation has been clinically shown to reverse the degenerative effects of neurodegenerative diseases such as Parkinson¿s and Alzheimer's Disease.Crucial Event Therapy describes medicine as the operational control of the functions of the human body treated as a network-of-networks, with 1/f-variable crucial events coding the dynamic states of health and disease through information flow within a network and information exchange between biomedical networks. A new way of thinking based on the statistics of Cortical Events is presented and the relevant literature is suitably referenced. This is anideal book for biophysicists and data scientists seeking to understand the connection of complexity measures for the study of consciousness with the clinical aspects of designing a rehabilitation strategy.

This book gathers peer-reviewed contributions submitted to the 21st European Conference on Mathematics for Industry, ECMI 2021, which was virtually held online, hosted by the University of Wuppertal, Germany, from April 13th to April 15th, 2021. The works explore mathematics in a wide variety of applications, ranging from problems in electronics, energy and the environment, to mechanics and mechatronics. Topics covered include: Applied Physics, Biology and Medicine, Cybersecurity, Data Science, Economics, Finance and Insurance, Energy, Production Systems, Social Challenges, and Vehicles and Transportation.The goal of the European Consortium for Mathematics in Industry (ECMI) conference series is to promote interaction between academia and industry, leading to innovations in both fields. These events have attracted leading experts from business, science and academia, and have promoted the application of novel mathematical technologies to industry. They have also encouraged industrial sectors to share challenging problems where mathematicians can provide fresh insights and perspectives. Lastly, the ECMI conferences are one of the main forums in which significant advances in industrial mathematics are presented, bringing together prominent figures from business, science and academia to promote the use of innovative mathematics in industry.