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In its simplest form, Hodge theory is the study of periods - integrals of algebraic differential forms which arise in the study of complex geometry and moduli, number theory and physics. Organized around the basic concepts of variations of Hodge structure and period maps, this volume draws together new developments in deformation theory, mirror symmetry, Galois representations, iterated integrals, algebraic cycles and the Hodge conjecture. Its mixture of high-quality expository and research articles make it a useful resource for graduate students and seasoned researchers alike.

Algebraic cycles --- Differential-algebraic equations --- Geometry, Algebraic --- Hodge theory --- Complex manifolds --- Differentiable manifolds --- Homology theory --- Algebraic-differential equations --- Differential-algebraic systems --- Equations, Algebraic-differential --- Equations, Differential-algebraic --- Systems, Differential-algebraic --- Differential equations --- Cycles, Algebraic

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The need for a rigorous mathematical theory for Differential-Algebraic Equations (DAEs) has its roots in the widespread applications of controlled dynamical systems, especially in mechanical and electrical engineering. Due to the strong relation to (ordinary) differential equations, the literature for DAEs mainly started out from introductory textbooks. As such, the present monograph is new in the sense that it comprises survey articles on various fields of DAEs, providing reviews, presentations of the current state of research and new concepts in -  Controllability for linear DAEs -  Port-Hamiltonian differential-algebraic systems -  Robustness of DAEs -  Solution concepts for DAEs -  DAEs in circuit modeling. The results in the individual chapters are presented in an accessible style, making this book suitable not only for active researchers but also for graduate students (with a good knowledge of the basic principles of DAEs) for self-study.

Differential-algebraic equations. --- Electric circuits -- Mathematical models. --- Mathematics. --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Differential equations. --- Algebraic-differential equations --- Differential-algebraic systems --- Equations, Algebraic-differential --- Equations, Differential-algebraic --- Systems, Differential-algebraic --- 517.91 Differential equations --- Differential equations --- System theory. --- Numerical analysis. --- Ordinary Differential Equations. --- Numerical Analysis. --- Systems Theory, Control. --- Differential Equations. --- Systems theory. --- Mathematical analysis --- Systems, Theory of --- Systems science --- Science --- Philosophy

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This textbook provides a rigorous and lucid introduction to the theory of ordinary differential equations (ODEs), which serve as mathematical models for many exciting real-world problems in science, engineering, and other disciplines. Key Features of this textbook: Effectively organizes the subject into easily manageable sections in the form of 42 class-tested lectures Provides a theoretical treatment by organizing the material around theorems and proofs Uses detailed examples to drive the presentation Includes numerous exercise sets that encourage pursuing extensions of the material, each with an "answers or hints" section Covers an array of advanced topics which allow for flexibility in developing the subject beyond the basics Provides excellent grounding and inspiration for future research contributions to the field of ODEs and related areas This book is ideal for a senior undergraduate or a graduate-level course on ordinary differential equations. Prerequisites include a course in calculus. Series: Universitext Ravi P. Agarwal received his Ph.D. in mathematics from the Indian Institute of Technology, Madras, India. He is a professor of mathematics at the Florida Institute of Technology. His research interests include numerical analysis, inequalities, fixed point theorems, and differential and difference equations. He is the author/co-author of over 800 journal articles and more than 20 books, and actively contributes to over 40 journals and book series in various capacities. Donal O’Regan received his Ph.D. in mathematics from Oregon State University, Oregon, U.S.A. He is a professor of mathematics at the National University of Ireland, Galway. He is the author/co-author of 14 books and has published over 650 papers on fixed point theory, operator, integral, differential and difference equations. He serves on the editorial board of many mathematical journals. Previously, the authors have co-authored/co-edited the following books with Springer: Infinite Interval Problems for Differential, Difference and Integral Equations; Singular Differential and Integral Equations with Applications; Nonlinear Analysis and Applications: To V. Lakshmikanthan on his 80th Birthday. In addition, they have collaborated with others on the following titles: Positive Solutions of Differential, Difference and Integral Equations; Oscillation Theory for Difference and Functional Differential Equations; Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations.

Mathematics. --- Ordinary Differential Equations. --- Partial Differential Equations. --- Appl.Mathematics/Computational Methods of Engineering. --- Differential Equations. --- Differential equations, partial. --- Engineering mathematics. --- Mathématiques --- Mathématiques de l'ingénieur --- Differential equations --- Differential equations. --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Differential-algebraric equations. --- 517.91 Differential equations --- Partial differential equations. --- Applied mathematics. --- Engineering --- Engineering analysis --- Mathematical analysis --- Partial differential equations --- Math --- Science --- Differential-algebraic equations. --- Algebraic-differential equations --- Differential-algebraic systems --- Equations, Algebraic-differential --- Equations, Differential-algebraic --- Systems, Differential-algebraic --- Mathematical and Computational Engineering.

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This book presents the various algebraic techniques for solving partial differential equations to yield exact solutions, techniques developed by the author in recent years and with emphasis on physical equations such as: the Maxwell equations, the Dirac equations, the KdV equation,  the KP equation,  the nonlinear Schrodinger equation,  the Davey and Stewartson equations, the Boussinesq equations in geophysics,  the Navier-Stokes equations and the boundary layer problems.  In order to solve them, I have employed the grading technique, matrix differential operators, stable-range of nonlinear terms, moving frames, asymmetric assumptions,  symmetry transformations,  linearization techniques  and  special functions. The book is self-contained and requires only a minimal understanding of calculus and linear algebra, making it accessible to a broad audience in the fields of mathematics, the sciences and engineering. Readers may find the exact solutions and mathematical skills needed in their own research.

Differential equations, Hyperbolic -- Numerical solutions. --- Differential equations, Partial -- Numerical solutions. --- Differential equations, Partial. --- Mathematics. --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Differential equations. --- Differential-algebraic equations. --- Algebraic-differential equations --- Differential-algebraic systems --- Equations, Algebraic-differential --- Equations, Differential-algebraic --- Systems, Differential-algebraic --- 517.91 Differential equations --- Differential equations --- Partial differential equations. --- Applied mathematics. --- Engineering mathematics. --- Mathematical physics. --- Partial Differential Equations. --- Mathematical Physics. --- Applications of Mathematics. --- Differential equations, partial. --- Partial differential equations --- Math --- Science --- Engineering --- Engineering analysis --- Mathematical analysis --- Physical mathematics --- Physics --- Differential equations, Partial --- Numerical solutions.

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Differential algebraic equations (DAEs), including so-called descriptor systems, began to attract significant research interest in applied and numerical mathematics in the early 1980s, no more than about three decades ago. In this relatively short time, DAEs have become a widely acknowledged tool to model processes subjected to certain constraints in order to simulate and to control processes in various application fields such as network simulation, chemical kinematics, mechanical engineering and systems biology. DAEs and their more abstract versions in infinite dimensional spaces comprise a great potential for the future mathematical modeling of complex coupled processes. The purpose of the book is to expose the impressive complexity of general DAEs from an analytical point of view, to describe the state of the art as well as open problems and in so doing to motivate further research of this versatile, extraordinary topic from a broader mathematical perspective. The book elaborates on a new general, structural analysis capturing linear and nonlinear DAEs in a hierarchical way. The DAE structure is exposed by means of special projector functions. Some issues on numerical integration and computational aspects are also treated in this context. .

Differential equations. --- Differential-algebraic equations. --- Electric circuits -- Mathematical models. --- Quantum groups. --- Differential-algebraic equations --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Algebraic-differential equations --- Differential-algebraic systems --- Equations, Algebraic-differential --- Equations, Differential-algebraic --- Systems, Differential-algebraic --- Mathematics. --- Applied mathematics. --- Engineering mathematics. --- Computer mathematics. --- Ordinary Differential Equations. --- Applications of Mathematics. --- Computational Mathematics and Numerical Analysis. --- Differential equations --- Differential Equations. --- Computer science --- Computer mathematics --- Discrete mathematics --- Electronic data processing --- Math --- Science --- 517.91 Differential equations --- Engineering --- Engineering analysis --- Mathematical analysis