Mathematical Modelling sets out the general principles of mathematical modelling as a means comprehending the world. Within the book, the problems of physics, engineering, chemistry, biology, medicine, economics, ecology, sociology, psychology, political science, etc. are all considered through this uniform lens. The author describes different classes of models, including lumped and distributed parameter systems, deterministic and stochastic models, continuous and discrete models, static and dynamical systems, and more. From a mathematical point of view, the considered models can be understood as equations and systems of equations of different nature and variational principles. In addition to this, mathematical features of mathematical models, applied control and optimization problems based on mathematical models, and identification of mathematical models are also presented. Features Each chapter includes four levels: a lecture (main chapter material), an appendix (additional information), notes (explanations, technical calculations, literature review) and tasks for independent work; this is suitable for undergraduates and graduate students and does not require the reader to take any prerequisite course, but may be useful for researchers as well Described mathematical models are grouped both by areas of application and by the types of obtained mathematical problems, which contributes to both the breadth of coverage of the material and the depth of its understanding Can be used as the main textbook on a mathematical modelling course, and is also recommended for special courses on mathematical models for physics, chemistry, biology, economics, etc.
Optimization: 100 Examples is a book devoted to the analysis of scenarios for which the use of well-known optimization methods encounter certain difficulties. Analysing such examples allows a deeper understanding of the features of these optimization methods, including the limits of their applicability. In this way, the book seeks to stimulate further development and understanding of the theory of optimal control. The study of the presented examples makes it possible to more effectively diagnose problems that arise in the practical solution of optimal control problems, and to find ways to overcome the difficulties that have arisen. Features Vast collection of examples Simple. accessible presentation Suitable as a research reference for anyone with an interest in optimization and optimal control theory, including mathematicians and engineers Examples differ in properties, i.e. each effect for each class of problems is illustrated by a unique example. Simon Serovajsky is a professor of mathematics at Al-Farabi Kazakh National University in Kazakhstan. He is the author of many books published in the area of optimization and optimal control theory, mathematical physics, mathematical modelling, philosophy and history of mathematics as well as a long list of high-quality publications in learned journals.
Besides financial goals, companies increasingly start formulating strategic environmental goals. However, the management and control of these goals remains a challenge. Therefore, this thesis aims to develop a method that is able to systematically integrate financial and environmental data in investment processes to support the achievement of strategic environmental and financial goals of companies. This new integrated investing method intends to provide a sufficient degree of scientific quality on the one hand and practical applicability on the other hand. In context of this thesis, a deficit analysis assesses reasons why current methods of environmental management accounting have not established as common business practice so far. During the first part of the subsequent method development a set of seven requirements is composed. On basis of these requirements additional methods of environmental management and management accounting systems are evaluated resulting in a final set of three approaches representing the basis of developing the new integrated investing method. After developing and describing the new integrated investing method, the method is applied in seven case studies within the Volkswagen Group. This method application verifies the method's practical applicability as well as its ability to manage and control strategic environmental and financial goals. In conclusion, this dissertation contributes to the development and application of a new integrated investing method that aims to ensure a sufficient degree of scientific quality and practical applicability. With the help of this method, it is able to systematically integrate environmental impacts in investment processes of companies in order to manage and control the achievement of strategic environmental and financial goals.
Behavioural type systems in programming languages support the specification and verification of properties of programs beyond the traditional use of type systems to describe data processing. A major example of such a property is correctness of communication in concurrent and distributed systems, motivated by the importance of structured communication in modern software. Behavioural Types: from Theory to Tools presents programming languages and software tools produced by members of COST Action IC1201: Behavioural Types for Reliable Large-Scale Software Systems, a European research network that was funded from October 2012 to October 2016. As a survey of the most recent developments in the application of behavioural type systems, it is a valuable reference for researchers in the field, as well as an introduction to the area for graduate students and software developers.
The equations of mathematical physics are the mathematical models of the large class of phenomenon of physics, chemistry, biology, economics, etc. In Sequential Models of Mathematical Physics, the author considers the justification of the process of constructing mathematical models. The book seeks to determine the classic, generalized and sequential solutions, the relationship between these solutions, its direct physical sense, the methods of its practical finding, and its existence. Features Describes a sequential method based on the construction of space completion, as well as its applications in number theory, the theory of distributions, the theory of extremum, and mathematical physics Presentation of the material is carried out on the simplest example of a one-dimensional stationary heat transfer process; all necessary concepts and constructions are introduced and illustrated with elementary examples, which makes the material accessible to a wide area of readers The solution of a specific mathematical problem is obtained as a result of the joint application of methods and concepts from completely different mathematical directions
A clear and lucid bottom-up approach to the basic principles of evolutionary algorithms Evolutionary algorithms (EAs) are a type of artificial intelligence. EAs are motivated by optimization processes that we observe in nature, such as natural selection, species migration, bird swarms, human culture, and ant colonies. This book discusses the theory, history, mathematics, and programming of evolutionary optimization algorithms. Featured algorithms include genetic algorithms, genetic programming, ant colony optimization, particle swarm optimization, differential evolution, biogeography-based optimization, and many others. Evolutionary Optimization Algorithms: Provides a straightforward, bottom-up approach that assists the reader in obtaining a clear but theoretically rigorous understanding of evolutionary algorithms, with an emphasis on implementation Gives a careful treatment of recently developed EAs including opposition-based learning, artificial fish swarms, bacterial foraging, and many others and discusses their similarities and differences from more well-established EAs Includes chapter-end problems plus a solutions manual available online for instructors Offers simple examples that provide the reader with an intuitive understanding of the theory Features source code for the examples available on the author's website Provides advanced mathematical techniques for analyzing EAs, including Markov modeling and dynamic system modeling Evolutionary Optimization Algorithms: Biologically Inspired and Population-Based Approaches to Computer Intelligence is an ideal text for advanced undergraduate students, graduate students, and professionals involved in engineering and computer science.
Interactive Graphics for Data Analysis: Principles and Examples discusses exploratory data analysis (EDA) and how interactive graphical methods can help gain insights as well as generate new questions and hypotheses from datasets.Fundamentals of Interactive Statistical GraphicsThe first part of the book summarizes principles and methodology, demons
A Comprehensive Course in Analysis by Poincaré Prize winner Barry Simon is a five-volume set that can serve as a graduate-level analysis textbook with a lot of additional bonus information, including hundreds of problems and numerous notes that extend the text and provide important historical background. Depth and breadth of exposition make this set a valuable reference source for almost all areas of classical analysis. Part 1 is devoted to real analysis. From one point of view, it presents the infinitesimal calculus of the twentieth century with the ultimate integral calculus (measure theory) and the ultimate differential calculus (distribution theory). From another, it shows the triumph of abstract spaces: topological spaces, Banach and Hilbert spaces, measure spaces, Riesz spaces, Polish spaces, locally convex spaces, Fréchet spaces, Schwartz space, and spaces. Finally it is the study of big techniques, including the Fourier series and transform, dual spaces, the Baire category, fixed point theorems, probability ideas, and Hausdorff dimension. Applications include the constructions of nowhere differentiable functions, Brownian motion, space-filling curves, solutions of the moment problem, Haar measure, and equilibrium measures in potential theory.
Presenting theory while using Mathematica in a complementary way, Modern Differential Geometry of Curves and Surfaces with Mathematica, the third edition of Alfred Gray’s famous textbook, covers how to define and compute standard geometric functions using Mathematica for constructing new curves and surfaces from existing ones. Since Gray’s death, authors Abbena and Salamon have stepped in to bring the book up to date. While maintaining Gray's intuitive approach, they reorganized the material to provide a clearer division between the text and the Mathematica code and added a Mathematica notebook as an appendix to each chapter. They also address important new topics, such as quaternions. The approach of this book is at times more computational than is usual for a book on the subject. For example, Brioshi’s formula for the Gaussian curvature in terms of the first fundamental form can be too complicated for use in hand calculations, but Mathematica handles it easily, either through computations or through graphing curvature. Another part of Mathematica that can be used effectively in differential geometry is its special function library, where nonstandard spaces of constant curvature can be defined in terms of elliptic functions and then plotted. Using the techniques described in this book, readers will understand concepts geometrically, plotting curves and surfaces on a monitor and then printing them. Containing more than 300 illustrations, the book demonstrates how to use Mathematica to plot many interesting curves and surfaces. Including as many topics of the classical differential geometry and surfaces as possible, it highlights important theorems with many examples. It includes 300 miniprograms for computing and plotting various geometric objects, alleviating the drudgery of computing things such as the curvature and torsion of a curve in space.
Convexity is important in theoretical aspects of mathematics and also for economists and physicists. In this monograph the author provides a comprehensive insight into convex sets and functions including the infinite-dimensional case and emphasizing the analytic point of view. Chapter one introduces the reader to the basic definitions and ideas that play central roles throughout the book. The rest of the book is divided into four parts: convexity and topology on infinite-dimensional spaces; Loewner's theorem; extreme points of convex sets and related issues, including the Krein–Milman theorem and Choquet theory; and a discussion of convexity and inequalities. The connections between disparate topics are clearly explained, giving the reader a thorough understanding of how convexity is useful as an analytic tool. A final chapter overviews the subject's history and explores further some of the themes mentioned earlier. This is an excellent resource for anyone interested in this central topic.
The investigation of nonlinear dynamis in physical and engineering systems from the point of view of systems and control theory is important to develop better engineering systems. Synchronization of oscillators and output regulation for rigid body systems are two problem classes which are inherently nonlinear and are of great importance in applications. This thesis contains novel results for both problem classes. In the case of sychronization of oscillators we consider two different system classes and give sufficient or necessary conditions for synchronization. In the case of the output regulation problems for rigid body systems we provide a new two-step control design procedure, a detailed analysis for the error dynamics and an application scenario for satellite control. A highlight of the thesis is a new separation principle which is the underlying principle of the two-step design procedure for the output regulation problem.
Learn Business Intelligence Markup Language (Biml) for automating much of the repetitive, manual labor involved in data integration. We teach you how to build frameworks and use advanced Biml features to get more out of SQL Server Integration Services (SSIS), Transact-SQL (T-SQL), and SQL Server Analysis Services (SSAS) than you ever thought possible. The first part of the book starts with the basics—getting your development environment configured, Biml syntax, and scripting essentials. Whether a beginner or a seasoned Biml expert, the next part of the book guides you through the process of using Biml to build a framework that captures both your design patterns and execution management. Design patterns are reusable code blocks that standardize the approach you use to perform certain types of data integration, logging, and other key data functions. Design patterns solve common problems encountered when developing data integration solutions. Because you do not have to build the code from scratch each time, design patterns improve your efficiency as a Biml developer. In addition to leveraging design patterns in your framework, you will learn how to build a robust metadata store and how to package your framework into Biml bundles for deployment within your enterprise. In the last part of the book, we teach you more advanced Biml features and capabilities, such as SSAS development, T-SQL recipes, documentation autogeneration, and Biml troubleshooting. The Biml Book: Provides practical and applicable examples Teaches you how to use Biml to reduce development time while improving quality Takes you through solutions to common data integration and BI challenges What You'll Learn Master the basics of Business Intelligence Markup Language (Biml) Study patterns for automating SSIS package generation Build a Biml Framework Import and transform database schemas Automate generation of scripts and projects Who This Book Is For BI developers wishing to quickly locate previously tested solutions, Microsoft BI specialists, those seeking more information about solution automation and code generation, and practitioners of Data Integration Lifecycle Management (DILM) in the DevOps enterprise
As ubiquitous as the atmosphere, intelligent adaptive systems (IASs) surround us in our daily lives. When designed well, these systems sense users and their environments so that they can provide support in a manner that is not only responsive to the evolving situation, but unnoticed by the user. A synthesis of recent research and developments on IASs from the human factors (HF) and human–computer interaction (HCI) domains, Intelligent Adaptive Systems: An Interaction-Centered Design Perspective provides integrated design guidance and recommendations for researchers and system developers. The book explores a recognized lack of integration between the HF and HCI research communities, which has led to inconsistencies between the research approaches adopted, and a lack of exploitation of research from one field by the other. The authors integrate theories and methodologies from these domains to provide design recommendations for human–machine developers. They then establish design guidance through the review of conceptual frameworks, analytical methodologies, and design processes for intelligent adaptive systems. The book draws on case studies from the military, medical, and distance learning domains to illustrate intelligent system design to examine lessons learned. Outlining an interaction-centered perspective for designing an IAS, the book details methodologies for understanding human work in complex environments and offers understanding about why and how optimizing human–machine interaction should be central to the design of IASs. The authors present an analytical and design methodology as well as an implementation strategy that helps you choose the proper design framework for your needs.
Although there is no precise definition of a “fractal”, it is usually understood to be a set whose smaller parts, when magnified, resemble the whole. Self-similar and self-affine sets are those for which this resemblance is precise and given by a contracting similitude or affine transformation. The present book is devoted to this most basic class of fractal objects. The book contains both introductory material for beginners and more advanced topics, which continue to be the focus of active research. Among the latter are self-similar sets and measures with overlaps, including the much-studied infinite Bernoulli convolutions. Self-affine systems pose additional challenges; their study is often based on ergodic theory and dynamical systems methods. In the last twenty years there have been many breakthroughs in these fields, and our aim is to give introduction to some of them, often in the simplest nontrivial cases. The book is intended for a wide audience of mathematicians interested in fractal geometry, including students. Parts of the book can be used for graduate and even advanced undergraduate courses.
The ideas of this book originate from the mobile WAVE approach which allowed us, more than a half century ago, to implement citywide heterogeneous computer networks and solve distributed problems on them well before the internet. The invented paradigm evolved into Spatial Grasp Technology and resulted in a European patent and eight books. The volumes covered concrete applications in graph and network theory, defense and social systems, crisis management, simulation of global viruses, gestalt theory, collective robotics, space research, and related concepts. The obtained solutions often exhibited high system qualities like global integrity, distributed awareness, and even consciousness. This current book takes these important characteristics as primary research objectives, together with the theory of patterns covering them all. This book is oriented towards system scientists, application programmers, industry managers, defense and security commanders, and university students (especially those interested in advanced MSc and PhD projects on distributed system management), as well as philosophers, psychologists, and United Nations personnel.
Localization for underwater robots remains a challenging issue. Typical sensors, such as Global Navigation Satellite System (GNSS) receivers, cannot be used under the surface and other inertial systems suffer from a strong integration drift. On top of that, the seabed is generally uniform and unstructured, making it difficult to apply Simultaneous Localization and Mapping (SLAM) methods to perform localization. Reliable Robot Localization presents an innovative new method which can be characterized as a raw-data SLAM approach. It differs from extant methods by considering time as a standard variable to be estimated, thus raising new opportunities for state estimation, so far underexploited. However, such temporal resolution is not straightforward and requires a set of theoretical tools in order to achieve the main purpose of localization. This book not only presents original contributions to the field of mobile robotics, it also offers new perspectives on constraint programming and set-membership approaches. It provides a reliable contractor programming framework in order to build solvers for dynamical systems. This set of tools is illustrated throughout this book with realistic robotic applications.
The Spatial Grasp Model suggests uses beyond the theoretical, including the examination of hurricanes and forest fires. Investigating group behaviour of ocean animals, discovery of unknown terrain features, and path-findings in large transport networks truly demonstrates the real-world application of SGL.
An accessible textbook for students and practitioners of Atomic, Molecular, and Optical Physics. It will be useful for scientists working with lasers. The book comes with an extensive freely downloadable software package and many colourful and animated illustrations. Additional materials are available for instructors.
A Comprehensive Course in Analysis by Poincaré Prize winner Barry Simon is a five-volume set that can serve as a graduate-level analysis textbook with a lot of additional bonus information, including hundreds of problems and numerous notes that extend the text and provide important historical background. Depth and breadth of exposition make this set a valuable reference source for almost all areas of classical analysis. Part 2B provides a comprehensive look at a number of subjects of complex analysis not included in Part 2A. Presented in this volume are the theory of conformal metrics (including the Poincaré metric, the Ahlfors-Robinson proof of Picard's theorem, and Bell's proof of the Painlevé smoothness theorem), topics in analytic number theory (including Jacobi's two- and four-square theorems, the Dirichlet prime progression theorem, the prime number theorem, and the Hardy-Littlewood asymptotics for the number of partitions), the theory of Fuschian differential equations, asymptotic methods (including Euler's method, stationary phase, the saddle-point method, and the WKB method), univalent functions (including an introduction to SLE), and Nevanlinna theory. The chapters on Fuschian differential equations and on asymptotic methods can be viewed as a minicourse on the theory of special functions.
We live in the age of Computer Business Systems (CBSs) -- the highly complex, computer-intensive management programs on which large organizations increasingly rely. In Mindless, Simon Head argues that these systems have come to trump human expertise, dictating the goals and strategies of a wide array of businesses, and de-skilling the jobs of middle class workers in the process. CBSs are especially dysfunctional, Head argues, when they apply their disembodied expertise to transactions between humans, as in health care, education, customer relations, and human resources management. And yet there are industries with more human approaches, as Head illustrates with specific examples, whose lead we must follow and extend to the mainstream American economy. Mindless illustrates the shortcomings of CBS, providing an in-depth and disturbing look at how human dignity is slipping as we become cogs on a white collar assembly line.
In formal language theory, the Parikh-image describes the absolute frequencies of symbols in words of a given language. The Parikh-images of regular languages are the same as the ones of context-free languages. These kinds of sets are called semilinear. Another algebraically defined class of sets has played an important role since the early days of formal language theory: recognizable subsets of monoids are a generalization of regular languages. A set is recognizable if and only if its syntactic monoid is finite. The first part of this monograph gives new results on semilinear sets. The descriptional complexity of operations is investigated. Semirecognizable subsets of monoids are introduced. Semirecognizability demands that the projection of the subset to its syntactic monoid is finite. The semirecognizable subsets of finitely generated free commutative monoids, which form a proper subset of the semilinear sets, are studied. Connections to rational cones enable the use of geometric methods. Jumping finite automata are a model for discontinuous information processing that has attracted interest for some years. Their operational state complexity and a variant called right one-way jumping finite automata are explored in the second part. We show that a permutation closed language is accepted by this variant if and only if it is semirecognizable. Results from the first part are used to get a better insight into these devices.
Evolutionary computation algorithms are employed to minimize functions with large number of variables. Biogeography-based optimization (BBO) is an optimization algorithm that is based on the science of biogeography, which researches the migration patterns of species. These migration paradigms provide the main logic behind BBO. Due to the cross-disciplinary nature of the optimization problems, there is a need to develop multiple approaches to tackle them and to study the theoretical reasoning behind their performance. This book explains the mathematical model of BBO algorithm and its variants created to cope with continuous domain problems (with and without constraints) and combinatorial problems.
Bayesian methods are increasingly being used in the social sciences, as the problems encountered lend themselves so naturally to the subjective qualities of Bayesian methodology. This book provides an accessible introduction to Bayesian methods, tailored specifically for social science students. It contains lots of real examples from political science, psychology, sociology, and economics, exercises in all chapters, and detailed descriptions of all the key concepts, without assuming any background in statistics beyond a first course. It features examples of how to implement the methods using WinBUGS – the most-widely used Bayesian analysis software in the world – and R – an open-source statistical software. The book is supported by a Website featuring WinBUGS and R code, and data sets.
Leading international security expert Peter Sapaty introduces a new, high-level distributed processing and control approach capable of finding real-time solutions for irregularities, crises, and security problems emerging any time and in any part of the world.
Everything you need to know in order to manage risk effectively within your organization You cannot afford to ignore the explosion in mathematical finance in your quest to remain competitive. This exciting branch of mathematics has very direct practical implications: when a new model is tested and implemented it can have an immediate impact on the financial environment. With risk management top of the agenda for many organizations, this book is essential reading for getting to grips with the mathematical story behind the subject of financial risk management. It will take you on a journey—from the early ideas of risk quantification up to today's sophisticated models and approaches to business risk management. To help you investigate the most up-to-date, pioneering developments in modern risk management, the book presents statistical theories and shows you how to put statistical tools into action to investigate areas such as the design of mathematical models for financial volatility or calculating the value at risk for an investment portfolio. Respected academic author Simon Hubbert is the youngest director of a financial engineering program in the U.K. He brings his industry experience to his practical approach to risk analysis Captures the essential mathematical tools needed to explore many common risk management problems Website with model simulations and source code enables you to put models of risk management into practice Plunges into the world of high-risk finance and examines the crucial relationship between the risk and the potential reward of holding a portfolio of risky financial assets This book is your one-stop-shop for effective risk management.
This textbook is aimed at transitioning high-school students who have already developed proficiency in mathematical problem solving from numerical-answer problems to proof-based mathematics. It serves to guide students on how to write and understand mathematical proofs. It covers proof techniques that are commonly used in several areas of mathematics, especially number theory, combinatorics, and analysis. In addition to just teaching the mechanics of proofs, this book showcases key materials in these areas, thus introducing readers to interesting mathematics along with proof techniques.
This book presents a comprehensive overview of the sum rule approach to spectral analysis of orthogonal polynomials, which derives from Gábor Szego's classic 1915 theorem and its 1920 extension. Barry Simon emphasizes necessary and sufficient conditions, and provides mathematical background that until now has been available only in journals. Topics include background from the theory of meromorphic functions on hyperelliptic surfaces and the study of covering maps of the Riemann sphere with a finite number of slits removed. This allows for the first book-length treatment of orthogonal polynomials for measures supported on a finite number of intervals on the real line. In addition to the Szego and Killip-Simon theorems for orthogonal polynomials on the unit circle (OPUC) and orthogonal polynomials on the real line (OPRL), Simon covers Toda lattices, the moment problem, and Jacobi operators on the Bethe lattice. Recent work on applications of universality of the CD kernel to obtain detailed asymptotics on the fine structure of the zeros is also included. The book places special emphasis on OPRL, which makes it the essential companion volume to the author's earlier books on OPUC.
Mozart's greatest works were written in Vienna in the decade before his death (1781–1791). This biography focuses on Mozart's dual roles as a performer and composer and reveals how his compositional processes are affected by performance-related concerns. It traces consistencies and changes in Mozart's professional persona and his modus operandi and sheds light on other prominent musicians, audience expectations, publishing, and concert and dramatic practices and traditions. Giving particular prominence to primary sources, Simon P. Keefe offers new biographical and critical perspectives on the man and his music, highlighting his extraordinary ability to engage with the competing demands of singers and instrumentalists, publishing and public performance, and concerts and dramatic productions in the course of a hectic, diverse and financially uncertain freelance career. This comprehensive and accessible volume is essential for Mozart lovers and scholars alike, exploring his Viennese masterpieces and the people and environments that shaped them.
A Comprehensive Course in Analysis by Poincaré Prize winner Barry Simon is a five-volume set that can serve as a graduate-level analysis textbook with a lot of additional bonus information, including hundreds of problems and numerous notes that extend the text and provide important historical background. Depth and breadth of exposition make this set a valuable reference source for almost all areas of classical analysis. Part 4 focuses on operator theory, especially on a Hilbert space. Central topics are the spectral theorem, the theory of trace class and Fredholm determinants, and the study of unbounded self-adjoint operators. There is also an introduction to the theory of orthogonal polynomials and a long chapter on Banach algebras, including the commutative and non-commutative Gel'fand-Naimark theorems and Fourier analysis on general locally compact abelian groups.
Fundamentals of Ocean Renewable Energy: Generating Electricity from the Sea presents the basic concepts of mechanics and introduces the various technical aspects of ocean renewable energy. Contents follow a logical sequence, starting with hydrodynamics and then separately examining each conversion technology, with special focus on tidal energy, offshore wind and wave energy, as well as current and ocean thermal energy conversion (OTEC). The authors explore key topics for resource characterization and optimization, such as monitoring and measurement methods and ocean modeling. They also discuss the sustainability, planning, integration and distribution challenges for the implementation of these technologies, including co-location with other systems. Finally, case studies of ocean energy sites and devices allow for a better understanding of how ocean energy conversion works in real-world settings. This book is an invaluable resource for students at graduate and senior undergraduate level engineering (ocean, mechanical, and civil) and oceanography with prior knowledge of fluid mechanics and mechanics of materials. - Presents the fundamental physics and theory behind ocean energy systems, covering both oceanographic and engineering aspects of ocean energy - Explores the most widely adopted conversion technologies, including tidal, wave, offshore wind, ocean thermal and currents
This text is a comprehensive pedagogical presentation of the theory of representation of finite and compact Lie groups. It considers both the general theory and representation of specific groups. Representation theory is discussed on the following types of groups: finite groups of rotations, permutation groups, and classical compact semisimple Lie groups. Along the way, the structure theory of the compact semisimple Lie groups is exposed. This is aimed at research mathematicians and graduate students studying group theory.
This book is the first of a multivolume series devoted to an exposition of functional analysis methods in modern mathematical physics. It describes the fundamental principles of functional analysis and is essentially self-contained, although there are occasional references to later volumes. We have included a few applications when we thought that they would provide motivation for the reader. Later volumes describe various advanced topics in functional analysis and give numerous applications in classical physics, modern physics, and partial differential equations.
This book draws a colorful and widespread picture of global affine hypersurface theory up to the most recent state. Moreover, the recent development revealed that affine differential geometry – as differential geometry in general – has an exciting intersection area with other fields of interest, like partial differential equations, global analysis, convex geometry and Riemann surfaces. The second edition of this monograph leads the reader from introductory concepts to recent research. Since the publication of the first edition in 1993 there appeared important new contributions, like the solutions of two different affine Bernstein conjectures, due to Chern and Calabi, respectively. Moreover, a large subclass of hyperbolic affine spheres were classified in recent years, namely the locally strongly convex Blaschke hypersurfaces that have parallel cubic form with respect to the Levi-Civita connection of the Blaschke metric. The authors of this book present such results and new methods of proof.
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