Conformal Maps And Geometry

Author: Dmitry Beliaev
Editor: World Scientific
ISBN: 178634615X
File Size: 60,94 MB
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'I very much enjoyed reading this book … Each chapter comes with well thought-out exercises, solutions to which are given at the end of the chapter. Conformal Maps and Geometry presents key topics in geometric function theory and the theory of univalent functions, and also prepares the reader to progress to study the SLE. It succeeds admirably on both counts.'MathSciNetGeometric function theory is one of the most interesting parts of complex analysis, an area that has become increasingly relevant as a key feature in the theory of Schramm-Loewner evolution.Though Riemann mapping theorem is frequently explored, there are few texts that discuss general theory of univalent maps, conformal invariants, and Loewner evolution. This textbook provides an accessible foundation of the theory of conformal maps and their connections with geometry.It offers a unique view of the field, as it is one of the first to discuss general theory of univalent maps at a graduate level, while introducing more complex theories of conformal invariants and extremal lengths. Conformal Maps and Geometry is an ideal resource for graduate courses in Complex Analysis or as an analytic prerequisite to study the theory of Schramm-Loewner evolution.

Computational Conformal Geometry

Author: Xianfeng David Gu
Editor: International Pressof Boston Incorporated
ISBN:
File Size: 27,46 MB
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Computational conformal geometry is an emerging inter-disciplinary field, with applications to algebraic topology, differential geometry and Riemann surface theories applied to geometric modeling, computer graphics, computer vision, medical imaging, visualization, scientific computation, and many other engineering fields.This new volume presents thorough introductions to the theoretical foundations—as well as to the practical algorithms—of computational conformal geometry. These have direct applications to engineering and digital geometric processing, including surface parameterization, surface matching, brain mapping, 3-D face recognition and identification, facial expression and animation, dynamic face tracking, mesh-spline conversion, and more.

Construction And Applications Of Conformal Maps

Author: Institute for Numerical Analysis (U.S.)
Editor:
ISBN:
File Size: 25,13 MB
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Riemannian Geometry

Author: Source Wikipedia
Editor: University-Press.org
ISBN: 9781230583549
File Size: 80,78 MB
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Pages: 151. Chapters: Nash embedding theorem, Conformal map, Curvature, Geodesic, Riemannian manifold, Riemann curvature tensor, Metric tensor, Holonomy, Symmetric space, De Sitter invariant special relativity, Covariance and contravariance of vectors, Geometrization conjecture, Systolic geometry, Covariant derivative, Hodge dual, Einstein notation, Ricci curvature, Ricci flow, Spin structure, Exponential map, List of formulas in Riemannian geometry, Introduction to systolic geometry, Christoffel symbols, Laplace-Beltrami operator, Glossary of Riemannian and metric geometry, Parallel transport, Gauss-Codazzi equations, Curvature of Riemannian manifolds, Gauss's lemma, Uniformization theorem, Levi-Civita connection, Ricci decomposition, Second fundamental form, Sectional curvature, De Sitter-Schwarzschild metric, Poincare metric, Calibrated geometry, Gravitational instanton, De Sitter space, Calculus of moving surfaces, Hermitian manifold, Tortuosity, Weyl tensor, Scalar curvature, Harmonic map, Cartan-Hadamard theorem, Pseudo-Riemannian manifold, Lie bracket of vector fields, Hermitian symmetric space, Spherical 3-manifold, Geodesics as Hamiltonian flows, Kahler manifold, Abel-Jacobi map, Jacobi field, Constraint counting, Clifford bundle, Killing vector field, Normal coordinates, Systoles of surfaces, Curved space, Fundamental theorem of Riemannian geometry, Theorema Egregium, Hyperkahler manifold, Unit tangent bundle, Sasakian manifold, Loewner's torus inequality, Cartan-Karlhede algorithm, Pu's inequality, Gauss map, Einstein manifold, Recurrent tensor, Soul theorem, Harmonic coordinates, Ruppeiner geometry, Filling radius, G2 manifold, Sub-Riemannian manifold, Quaternion-Kahler manifold, Frobenius manifold, Gromov-Hausdorff convergence, Cheng's eigenvalue comparison theorem, Gromov's systolic inequality for...

Conformal Mapping

Author: Ludwig Bieberbach
Editor: American Mathematical Soc.
ISBN: 0821821059
File Size: 29,11 MB
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Translated from the fourth German edition by F. Steinhardt, with an expanded Bibliography.

Inversion Theory And Conformal Mapping

Author: David E. Blair
Editor: American Mathematical Soc.
ISBN: 0821826360
File Size: 26,16 MB
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It is rarely taught in an undergraduate or even graduate curriculum that the only conformal maps in Euclidean space of dimension greater than two are those generated by similarities and inversions in spheres. This is in stark contrast to the wealth of conformal maps in the plane. The principal aim of this text is to give a treatment of this paucity of conformal maps in higher dimensions. The exposition includes both an analytic proof in general dimension and a differential-geometric proof in dimension three. For completeness, enough complex analysis is developed to prove the abundance of conformal maps in the plane. In addition, the book develops inversion theory as a subject, along with the auxiliary theme of circle-preserving maps. A particular feature is the inclusion of a paper by Caratheodory with the remarkable result that any circle-preserving transformation is necessarily a Mobius transformation, not even the continuity of the transformation is assumed. The text is at the level of advanced undergraduates and is suitable for a capstone course, topics course, senior seminar or independent study. Students and readers with university courses in differential geometry or complex analysis bring with them background to build on, but such courses are not essential prerequisites.

The Theory And Practice Of Conformal Geometry

Author: Steven G. Krantz
Editor: Courier Dover Publications
ISBN: 0486793443
File Size: 19,77 MB
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An expert on conformal geometry introduces some of the subject's modern developments. Topics include the Riemann mapping theorem, invariant metrics, automorphism groups, harmonic measure, extremal length, analytic capacity, invariant geometry, and more. 2016 edition.

Quasiconformal Maps And Teichm Ller Theory

Author: Alastair Fletcher
Editor: Oxford University Press on Demand
ISBN:
File Size: 64,37 MB
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Handbook Of Complex Analysis

Author: Reiner Kuhnau
Editor: Elsevier
ISBN: 9780080495170
File Size: 44,30 MB
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Geometric Function Theory is that part of Complex Analysis which covers the theory of conformal and quasiconformal mappings. Beginning with the classical Riemann mapping theorem, there is a lot of existence theorems for canonical conformal mappings. On the other side there is an extensive theory of qualitative properties of conformal and quasiconformal mappings, concerning mainly a prior estimates, so called distortion theorems (including the Bieberbach conjecture with the proof of the Branges). Here a starting point was the classical Scharz lemma, and then Koebe's distortion theorem. There are several connections to mathematical physics, because of the relations to potential theory (in the plane). The Handbook of Geometric Function Theory contains also an article about constructive methods and further a Bibliography including applications eg: to electroxtatic problems, heat conduction, potential flows (in the plane). · A collection of independent survey articles in the field of GeometricFunction Theory · Existence theorems and qualitative properties of conformal and quasiconformal mappings · A bibliography, including many hints to applications in electrostatics, heat conduction, potential flows (in the plane).

On Brennan S Conjecture In Conformal Mapping

Author: Daniel Bertilsson
Editor:
ISBN:
File Size: 55,87 MB
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Handbook Of Conformal Mappings And Applications

Author: Prem K. Kythe
Editor: CRC Press
ISBN: 1351718738
File Size: 11,19 MB
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The subject of conformal mappings is a major part of geometric function theory that gained prominence after the publication of the Riemann mapping theorem — for every simply connected domain of the extended complex plane there is a univalent and meromorphic function that maps such a domain conformally onto the unit disk. The Handbook of Conformal Mappings and Applications is a compendium of at least all known conformal maps to date, with diagrams and description, and all possible applications in different scientific disciplines, such as: fluid flows, heat transfer, acoustics, electromagnetic fields as static fields in electricity and magnetism, various mathematical models and methods, including solutions of certain integral equations.

Geometric Geodesy 3d Geodesy Conformal Mapping

Author: Alfred Leick
Editor:
ISBN:
File Size: 24,54 MB
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Two Dimensional Conformal Geometry And Vertex Operator Algebras

Author: Yi-Zhi Huang
Editor: Springer Science & Business Media
ISBN: 1461242762
File Size: 59,69 MB
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The theory of vertex operator algebras and their representations has been showing its power in the solution of concrete mathematical problems and in the understanding of conceptual but subtle mathematical and physical struc- tures of conformal field theories. Much of the recent progress has deep connec- tions with complex analysis and conformal geometry. Future developments, especially constructions and studies of higher-genus theories, will need a solid geometric theory of vertex operator algebras. Back in 1986, Manin already observed in Man) that the quantum theory of (super )strings existed (in some sense) in two entirely different mathematical fields. Under canonical quantization this theory appeared to a mathematician as the representation theories of the Heisenberg, Vir as oro and affine Kac- Moody algebras and their superextensions. Quantization with the help of the Polyakov path integral led on the other hand to the analytic theory of algebraic (super ) curves and their moduli spaces, to invariants of the type of the analytic curvature, and so on.He pointed out further that establishing direct mathematical connections between these two forms of a single theory was a big and important problem. On the one hand, the theory of vertex operator algebras and their repre- sentations unifies (and considerably extends) the representation theories of the Heisenberg, Virasoro and Kac-Moody algebras and their superextensions.

Geometric Theory Of Functions Of A Complex Variable

Author: Gennadiĭ Mikhaĭlovich Goluzin
Editor: American Mathematical Soc.
ISBN: 9780821886557
File Size: 52,32 MB
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Conformal Geometry And Quasiregular Mappings

Author: Matti Vuorinen
Editor: Springer
ISBN: 3540392076
File Size: 27,29 MB
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This book is an introduction to the theory of spatial quasiregular mappings intended for the uninitiated reader. At the same time the book also addresses specialists in classical analysis and, in particular, geometric function theory. The text leads the reader to the frontier of current research and covers some most recent developments in the subject, previously scatterd through the literature. A major role in this monograph is played by certain conformal invariants which are solutions of extremal problems related to extremal lengths of curve families. These invariants are then applied to prove sharp distortion theorems for quasiregular mappings. One of these extremal problems of conformal geometry generalizes a classical two-dimensional problem of O. Teichmüller. The novel feature of the exposition is the way in which conformal invariants are applied and the sharp results obtained should be of considerable interest even in the two-dimensional particular case. This book combines the features of a textbook and of a research monograph: it is the first introduction to the subject available in English, contains nearly a hundred exercises, a survey of the subject as well as an extensive bibliography and, finally, a list of open problems.

Elliptic Partial Differential Equations And Quasiconformal Mappings In The Plane Pms 48

Author: Kari Astala
Editor: Princeton University Press
ISBN: 9780691137773
File Size: 41,19 MB
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This book explores the most recent developments in the theory of planar quasiconformal mappings with a particular focus on the interactions with partial differential equations and nonlinear analysis. It gives a thorough and modern approach to the classical theory and presents important and compelling applications across a spectrum of mathematics: dynamical systems, singular integral operators, inverse problems, the geometry of mappings, and the calculus of variations. It also gives an account of recent advances in harmonic analysis and their applications in the geometric theory of mappings. The book explains that the existence, regularity, and singular set structures for second-order divergence-type equations--the most important class of PDEs in applications--are determined by the mathematics underpinning the geometry, structure, and dimension of fractal sets; moduli spaces of Riemann surfaces; and conformal dynamical systems. These topics are inextricably linked by the theory of quasiconformal mappings. Further, the interplay between them allows the authors to extend classical results to more general settings for wider applicability, providing new and often optimal answers to questions of existence, regularity, and geometric properties of solutions to nonlinear systems in both elliptic and degenerate elliptic settings.

Conformal Invariants Inequalities And Quasiconformal Maps

Author: Glen D. Anderson
Editor: Wiley-Interscience
ISBN:
File Size: 22,77 MB
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Disk contains: information on Conformal Invariants Software which accompanies the text.

The Cauchy Transform Potential Theory And Conformal Mapping

Author: Steven R. Bell
Editor: CRC Press
ISBN: 9780849382703
File Size: 80,61 MB
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The Cauchy integral formula is the most central result in all of classical function theory. A recent discovery of Kerzman and Stein allows more theorems than ever to be deduced from simple facts about the Cauchy integral. In this book, the Riemann Mapping Theorem is deduced, the Dirichlet and Neumann problems for the Laplace operator are solved, the Poisson kernal is constructed, and the inhomogenous Cauchy-Reimann equations are solved concretely using formulas stemming from the Kerzman-Stein result. These explicit formulas yield new numerical methods for computing the classical objects of potential theory and conformal mapping, and the book provides succinct, complete explanations of these methods. The Cauchy Transform, Potential Theory, and Conformal Mapping is suitable for pure and applied math students taking a beginning graduate-level topics course on aspects of complex analysis. It will also be useful to physicists and engineers interested in a clear exposition on a fundamental topic of complex analysis, methods, and their application.

Geometric Function Theory

Author: Steven G. Krantz
Editor: Springer Science & Business Media
ISBN: 0817644407
File Size: 24,62 MB
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* Presented from a geometric analytical viewpoint, this work addresses advanced topics in complex analysis that verge on modern areas of research * Methodically designed with individual chapters containing a rich collection of exercises, examples, and illustrations

Complex Analysis

Author: Steven G. Krantz
Editor: Cambridge University Press
ISBN: 9780883850350
File Size: 55,67 MB
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In this second edition of a Carus Monograph Classic, Steven Krantz develops material on classical non-Euclidean geometry. He shows how it can be developed in a natural way from the invariant geometry of the complex disc. He also introduces the Bergman kernel and metric and provides profound applications, some of them never having appeared before in print. In general, the new edition represents a considerable polishing and re-thinking of the original successful volume. This is the first and only book to describe the context, the background, the details, and the applications of Ahlfors's celebrated ideas about curvature, the Schwarz lemma, and applications in complex analysis. Beginning from scratch, and requiring only a minimal background in complex variable theory, this book takes the reader up to ideas that are currently active areas of study. Such areas include a) the Caratheodory and Kobayashi metrics, b) the Bergman kernel and metric, c) boundary continuation of conformal maps. There is also an introduction to the theory of several complex variables. Poincar 's celebrated theorem about the biholomorphic inequivalence of the ball and polydisc is discussed and proved.