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Around 1994 R. Borcherds discovered a new type of meromorphic modular form on the orthogonal group $O(2,n)$. These "Borcherds products" have infinite product expansions analogous to the Dedekind eta-function. They arise as multiplicative liftings of elliptic modular forms on $(SL)_2(R)$. The fact that the zeros and poles of Borcherds products are explicitly given in terms of Heegner divisors makes them interesting for geometric and arithmetic applications. In the present text the Borcherds' construction is extended to Maass wave forms and is used to study the Chern classes of Heegner divisors. A converse theorem for the lifting is proved.

Forms, Modular. --- Series, Theta. --- Picard groups. --- Algebraic cycles. --- Chern classes. --- Forms, Modular --- Series, Theta --- Picard groups --- Algebraic cycles --- Chern classes --- Mathematical Theory --- Algebra --- Mathematics --- Physical Sciences & Mathematics --- Algebra. --- Field theory (Physics). --- Algebraic geometry. --- Field Theory and Polynomials. --- Algebraic Geometry. --- Algebraic geometry --- Geometry --- Classical field theory --- Continuum physics --- Physics --- Continuum mechanics --- Mathematical analysis

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This work treats an introduction to commutative ring theory and algebraic plane curves, requiring of the student only a basic knowledge of algebra, with all of the algebraic facts collected into several appendices that can be easily referred to, as needed. Kunz's proven conception of teaching topics in commutative algebra together with their applications to algebraic geometry makes this book significantly different from others on plane algebraic curves. The exposition focuses on the purely algebraic aspects of plane curve theory, leaving the topological and analytical viewpoints in the background, with only casual references to these subjects and suggestions for further reading. Most important to this text: * Emphasizes and utilizes the theory of filtered algebras, their graduated rings and Rees algebras, to deduce basic facts about the intersection theory of plane curves * Presents residue theory in the affine plane and its applications to intersection theory * Methods of proof for the Riemann–Roch theorem conform to the presentation of curve theory, formulated in the language of filtrations and associated graded rings * Examples, exercises, figures and suggestions for further study round out this fairly self-contained textbook From a review of the German edition: "[T]he reader is invited to learn some topics from commutative ring theory by mainly studying their illustrations and applications in plane curve theory. This methodical approach is certainly very enlightening and efficient for both teachers and students… The whole text is a real masterpiece of clarity, rigor, comprehension, methodical skill, algebraic and geometric motivation…highly enlightening, motivating and entertaining at the same time… One simply cannot do better in writing such a textbook." —Zentralblatt MATH .

Curves, Plane. --- Curves, Algebraic. --- Singularities (Mathematics) --- Algebraic curves --- Algebraic varieties --- Higher plane curves --- Plane curves --- Geometry, Algebraic --- Algebraic geometry --- Geometry, algebraic. --- Algebraic topology. --- Mathematics. --- Algebra. --- Field theory (Physics). --- Algebraic Geometry. --- Algebraic Topology. --- Applications of Mathematics. --- Commutative Rings and Algebras. --- Associative Rings and Algebras. --- Field Theory and Polynomials. --- Geometry --- Classical field theory --- Continuum physics --- Physics --- Continuum mechanics --- Mathematics --- Mathematical analysis --- Math --- Science --- Topology --- Algebraic geometry. --- Applied mathematics. --- Engineering mathematics. --- Commutative algebra. --- Commutative rings. --- Associative rings. --- Rings (Algebra). --- Algebra --- Engineering --- Engineering analysis --- Algebraic rings --- Ring theory --- Algebraic fields --- Rings (Algebra) --- Geometry, Algebraic. --- Field theory (Physics)

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This volume collects three series of lectures on applications of the theory of Hamiltonian systems, contributed by some of the specialists in the field. The aim is to describe the state of the art for some interesting problems, such as the Hamiltonian theory for infinite-dimensional Hamiltonian systems, including KAM theory, the recent extensions of the theory of adiabatic invariants and the phenomena related to stability over exponentially long times of Nekhoroshev's theory. The books may serve as an excellent basis for young researchers, who will find here a complete and accurate exposition of recent original results and many hints for further investigation.

Hamiltonian systems --- Systèmes hamiltoniens --- Congresses. --- Congrès --- Hamiltonian systems. --- Mathematics. --- Differentiable dynamical systems. --- Differential equations, partial. --- Cell aggregation --- Thermodynamics. --- Dynamical Systems and Ergodic Theory. --- Partial Differential Equations. --- Manifolds and Cell Complexes (incl. Diff.Topology). --- Mechanics, Fluids, Thermodynamics. --- Mathematical Theory --- Geometry --- Calculus --- Mathematics --- Physical Sciences & Mathematics --- Hamiltonian dynamical systems --- Systems, Hamiltonian --- Aggregation, Cell --- Cell patterning --- Partial differential equations --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Math --- Dynamics. --- Ergodic theory. --- Partial differential equations. --- Manifolds (Mathematics). --- Complex manifolds. --- Classical and Continuum Physics. --- Science --- Analytic spaces --- Manifolds (Mathematics) --- Geometry, Differential --- Topology --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Cell interaction --- Microbial aggregation --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Continuum physics. --- Classical field theory --- Continuum physics --- Continuum mechanics --- Differentiable dynamical systems

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This is Volume 4 of the book series of the Body & Soul mathematics education reform program, and presents a unified new approach to computational simulation of turbulent flow starting from the general basis of calculus and linear algebra of Vol 1-3. The book puts the Body & Soul computational finite element methodology in the form of General Galerkin (G2), up against the challenge of computing turbulent solutions of the inviscid Euler equations and the Navier-Stokes equations with small viscosity. The book shows that direct application of G2 without any turbulence or wall modeling, allows reliable computation on a PC of mean value quantities of turbulent flow such as drag and lift. The power of G2 is demonstrated by resolving several classical scientific paradoxes of fluid flow and by uncovering secrets of flying, sailing, racing and ball sports. The book presents new aspects on both mathematics and computation of turbulent flow, and challenges established approaches. The book is directed to a wide audience of computational mathematicians fluid dynamicists and scientists. The G2 solver is available as part the free software project FEniCS at www.fenics.org. The book has a dedicated dynamic web page, including movies from a wide variety of simulations, at www.bodysoulmath.org. The book is focussed on incompressible flow, but opens to compressible flow continued in Vol 5 on thermodynamics. The authors are experts on computational mathematics and technology.

Mathematical analysis --- Hydromechanics --- Conception assistée par ordinateur --- Mathematics. --- Algebras, Linear. --- Differential equations. --- Differential equations, Partial. --- Calculus of variations. --- Numerical analysis. --- Mathematical optimization. --- Fluid mechanics. --- Continuum mechanics --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- Isoperimetrical problems --- Variations, Calculus of --- Partial differential equations --- 517.91 Differential equations --- Differential equations --- Linear algebra --- Algebra, Universal --- Generalized spaces --- Calculus of operations --- Line geometry --- Topology --- Math --- Science --- Differential geometry. Global analysis --- Fluid mechanics --- 517.91. --- Numerical solutions --- Computer mathematics. --- Physics. --- Continuum physics. --- Applied mathematics. --- Engineering mathematics. --- Computational Mathematics and Numerical Analysis. --- Engineering Fluid Dynamics. --- Classical Continuum Physics. --- Computational Science and Engineering. --- Mathematical Methods in Physics. --- Appl.Mathematics/Computational Methods of Engineering. --- Differential equations, Partial --- Mathematical models --- Numerical analysis --- Equations différentielles --- Equations aux dérivées partielles --- Modèles mathématiques --- Analyse numérique --- Data processing --- Computer-aided design --- Informatique --- EPUB-LIV-FT LIVMATHE SPRINGER-B --- Computer science --- Hydraulic engineering. --- Computer science. --- Mathematical physics. --- Classical and Continuum Physics. --- Mathematical and Computational Engineering. --- Physical mathematics --- Physics --- Informatics --- Engineering, Hydraulic --- Engineering --- Hydraulics --- Shore protection --- Computer mathematics --- Discrete mathematics --- Electronic data processing --- Engineering analysis --- Mathematics --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Classical field theory --- Continuum physics --- Algebras, Linear --- 517.91 --- Calculus of variations --- Mathematical optimization --- Field theory (Physics)