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Curves.. --- Magnetism.. --- Mathematical physics --- Physics --- Electricity --- Magnetics --- Calculus --- Conic sections --- Geometry, Analytic --- Geometry, Differential --- Geometry, Enumerative --- Mathematics --- Shapes

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Path integrals provide a powerful method for describing quantum phenomena. This book introduces the quantum mechanics of particles that move in curved space by employing path integrals and then using them to compute anomalies in quantum field theories. The authors start by deriving path integrals for particles moving in curved space and their supersymmetric generalizations. They then discuss the regularization schemes essential to constructing and computing these path integrals. This topic is used to introduce regularization and renormalization in quantum field theories in a wider context. These methods are then applied to discuss and calculate anomalies in quantum field theory. Such anomalies provide enormous constraints in the search for physical theories of elementary particles, quantum gravity and string theories. An advanced text for researchers and graduate students of quantum field theory and string theory, the first part is also a stand-alone introduction to path integrals in quantum mechanics.

Quantum field theory --- Path integrals --- 530.19 --- Integrals, Path --- Integrals --- Probabilities --- Quantum theory --- Statistical physics --- Relativistic quantum field theory --- Field theory (Physics) --- Relativity (Physics) --- Fundamental functions in general. Potential. Gradient. Intensity. Capacity etc. --- Curves. --- Path integrals. --- 530.19 Fundamental functions in general. Potential. Gradient. Intensity. Capacity etc. --- Curves --- Calculus --- Conic sections --- Geometry, Analytic --- Geometry, Differential --- Geometry, Enumerative --- Mathematics --- Shapes --- Fundamental functions in general. Potential. Gradient. Intensity. Capacity etc

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This volume presents the invited lectures of the workshop "Infinite Dimensional Algebras and Quantum Integrable Systems'' held in July 2003 at the University of Algarve, Faro, Portugal, as a satellite workshop of the XIV. International Congress on Mathematical Physics. Recent developments in the theory of infinite dimensional algebras and their applications to quantum integrable systems are reviewed by some of the leading experts in the field. The volume will be of interest to a broad audience from graduate students to researchers in mathematical physics and related fields. Contributors: E. Frenkel O.A. Castro-Alvaredo and A. Fring V.G. Kac and M. Wakimoto A. Gerasimov, S. Kharchev and D. Lebedev H.E. Boos, V.E. Korepin and F.A. Smirnov Kanehisa Takasaki Takashi Takebe L.A. Takhtajan and Lee-Peng Teo V. Tarasov.

Lie algebras --- Lie superalgebras --- Curves --- Functions of several complex variables --- Quantum theory --- Calculus --- Conic sections --- Geometry, Analytic --- Geometry, Differential --- Geometry, Enumerative --- Mathematics --- Shapes --- Superalgebras --- Algebra. --- Quantum theory. --- Mathematical physics. --- Topological Groups. --- Matrix theory. --- Systems theory. --- Quantum Physics. --- Mathematical Methods in Physics. --- Topological Groups, Lie Groups. --- Linear and Multilinear Algebras, Matrix Theory. --- Systems Theory, Control. --- System theory. --- Groups, Topological --- Continuous groups --- Physical mathematics --- Physics --- Quantum dynamics --- Quantum mechanics --- Quantum physics --- Mechanics --- Thermodynamics --- Mathematical analysis --- Systems, Theory of --- Systems science --- Science --- Philosophy --- Quantum physics. --- Physics. --- Topological groups. --- Lie groups. --- Groups, Lie --- Symmetric spaces --- Topological groups --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics

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The book provides an introduction to Differential Geometry of Curves and Surfaces. The theory of curves starts with a discussion of possible definitions of the concept of curve, proving in particular the classification of 1-dimensional manifolds. We then present the classical local theory of parametrized plane and space curves (curves in n-dimensional space are discussed in the complementary material): curvature, torsion, Frenet’s formulas and the fundamental theorem of the local theory of curves. Then, after a self-contained presentation of degree theory for continuous self-maps of the circumference, we study the global theory of plane curves, introducing winding and rotation numbers, and proving the Jordan curve theorem for curves of class C2, and Hopf theorem on the rotation number of closed simple curves. The local theory of surfaces begins with a comparison of the concept of parametrized (i.e., immersed) surface with the concept of regular (i.e., embedded) surface. We then develop the basic differential geometry of surfaces in R3: definitions, examples, differentiable maps and functions, tangent vectors (presented both as vectors tangent to curves in the surface and as derivations on germs of differentiable functions; we shall consistently use both approaches in the whole book) and orientation. Next we study the several notions of curvature on a surface, stressing both the geometrical meaning of the objects introduced and the algebraic/analytical methods needed to study them via the Gauss map, up to the proof of Gauss’ Teorema Egregium. Then we introduce vector fields on a surface (flow, first integrals, integral curves) and geodesics (definition, basic properties, geodesic curvature, and, in the complementary material, a full proof of minimizing properties of geodesics and of the Hopf-Rinow theorem for surfaces). Then we shall present a proof of the celebrated Gauss-Bonnet theorem, both in its local and in its global form, using basic properties (fully proved in the complementary material) of triangulations of surfaces. As an application, we shall prove the Poincaré-Hopf theorem on zeroes of vector fields. Finally, the last chapter will be devoted to several important results on the global theory of surfaces, like for instance the characterization of surfaces with constant Gaussian curvature, and the orientability of compact surfaces in R3.

Curves -- Textbooks. --- Geometry, Differential. --- Surfaces -- Textbooks. --- Mathematics --- Physical Sciences & Mathematics --- Geometry --- Mathematical Theory --- Geometry, Differential --- Curves --- Surfaces --- Curved surfaces --- Differential geometry --- Mathematics. --- Computer graphics. --- Computer mathematics. --- Geometry. --- Differential geometry. --- Mathematics, general. --- Differential Geometry. --- Computational Science and Engineering. --- Computer Imaging, Vision, Pattern Recognition and Graphics. --- Shapes --- Calculus --- Conic sections --- Geometry, Analytic --- Geometry, Enumerative --- Global differential geometry. --- Computer science. --- Computer vision. --- Machine vision --- Vision, Computer --- Artificial intelligence --- Image processing --- Pattern recognition systems --- Informatics --- Science --- Euclid's Elements --- Math --- Curves. --- Surfaces. --- Optical data processing. --- Optical computing --- Visual data processing --- Bionics --- Electronic data processing --- Integrated optics --- Photonics --- Computers --- Computer mathematics --- Optical equipment

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Interactive curve modeling techniques and their applications are extremely useful in a number of academic and industrial settings, and specifically play a significant role in multidisciplinary problem solving, such as in font design, designing objects, CAD/CAM, medical operations, scientific data visualization, virtual reality, character recognition, and object recognition, etc. Various problems such as iris, fingerprint, and signature recognition, can also be intelligently solved and automated using curve techniques. This broad-ranging textbook covers curve modeling with solutions to real life problems relating to computer graphics, vision, image processing, geometric modeling and CAD/CAM. Well-explained, easy-to-understand chapters deal with basic concepts, curve design techniques and their use to various applications, and a wide range of problems with their automated solutions via computers. Features and topics: • Provides a class of practical solutions to real life and multidisciplinary problems • Offers students supporting pedagogical tools in the form of a thorough introductory chapter, individual chapter introductions and end summaries, as well as end-of-chapter exercises • Presents both classical and up-to-date theory, with practice to get problems solved in diverse disciplines • Focuses on interdisciplinary methods and up-to-date methodologies in the field • Imparts a description and analysis of a variety of classes of splines for use in CAGD (computer-aided geometric design), CAD (computer-aided design), CAE (computer-aided engineering), computer graphics, computer vision, image processing and other disciplines • Aims to stimulate views and provide a source where readers can find the latest state-of-the-art developments in the field, including a variety of techniques, applications, and systems necessary for solving problems Interactive Curve Modeling also will serve as an important tool for readers; as an extremely useful textbook for senior undergraduates as well as graduate students in the areas of computer science, engineering, and other computational sciences. This comprehensive text can equally act as an invaluable resource for those practitioners and researchers looking for an introduction to the state-of-the-art on the topic. Professor Sarfraz has many years of experience researching and teaching in the field, winning an award for Excellence in Research at the King Fahd University of Petroleum and Minerals, Saudi Arabia.

Curves. --- Geometry, Plane. --- Plane geometry --- Calculus --- Conic sections --- Geometry, Analytic --- Geometry, Differential --- Geometry, Enumerative --- Mathematics --- Shapes --- Computer vision. --- Computer graphics. --- Optical pattern recognition. --- Computer aided design. --- Image Processing and Computer Vision. --- Computer Graphics. --- Pattern Recognition. --- Computer-Aided Engineering (CAD, CAE) and Design. --- CAD (Computer-aided design) --- Computer-assisted design --- Computer-aided engineering --- Design --- Optical data processing --- Pattern perception --- Perceptrons --- Visual discrimination --- Automatic drafting --- Graphic data processing --- Graphics, Computer --- Computer art --- Graphic arts --- Electronic data processing --- Engineering graphics --- Image processing --- Machine vision --- Vision, Computer --- Artificial intelligence --- Pattern recognition systems --- Digital techniques --- Optical data processing. --- Pattern recognition. --- Computer-aided engineering. --- CAE --- Engineering --- Design perception --- Pattern recognition --- Form perception --- Perception --- Figure-ground perception --- Optical computing --- Visual data processing --- Bionics --- Integrated optics --- Photonics --- Computers --- Data processing --- Optical equipment