An Excursion Through Discrete Differential Geometry

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An Excursion Through Discrete Differential Geometry

American Mathematical Society. Short Course, Discrete Differential Geometry
An Excursion Through Discrete Differential Geometry Book PDF

Author : American Mathematical Society. Short Course, Discrete Differential Geometry
Publisher : American Mathematical Soc.
Page : 140 pages
File Size : 53,9 Mb
Release : 2020-09-02
Category : Education
ISBN : 9781470446628

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An Excursion Through Discrete Differential Geometry by American Mathematical Society. Short Course, Discrete Differential Geometry Pdf

Discrete Differential Geometry (DDG) is an emerging discipline at the boundary between mathematics and computer science. It aims to translate concepts from classical differential geometry into a language that is purely finite and discrete, and can hence be used by algorithms to reason about geometric data. In contrast to standard numerical approximation, the central philosophy of DDG is to faithfully and exactly preserve key invariants of geometric objects at the discrete level. This process of translation from smooth to discrete helps to both illuminate the fundamental meaning behind geometric ideas and provide useful algorithmic guarantees. This volume is based on lectures delivered at the 2018 AMS Short Course ``Discrete Differential Geometry,'' held January 8-9, 2018, in San Diego, California. The papers in this volume illustrate the principles of DDG via several recent topics: discrete nets, discrete differential operators, discrete mappings, discrete conformal geometry, and discrete optimal transport.

Discrete Differential Geometry

Alexander I. Bobenko TU Berlin,Peter Schröder,John M. Sullivan,Günter M. Ziegler
Discrete Differential Geometry Book PDF

Author : Alexander I. Bobenko TU Berlin,Peter Schröder,John M. Sullivan,Günter M. Ziegler
Publisher : Springer Science & Business Media
Page : 341 pages
File Size : 53,9 Mb
Release : 2008-03-27
Category : Mathematics
ISBN : 9783764386214

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Discrete Differential Geometry by Alexander I. Bobenko TU Berlin,Peter Schröder,John M. Sullivan,Günter M. Ziegler Pdf

This is the first book on a newly emerging field of discrete differential geometry providing an excellent way to access this exciting area. It provides discrete equivalents of the geometric notions and methods of differential geometry, such as notions of curvature and integrability for polyhedral surfaces. The carefully edited collection of essays gives a lively, multi-facetted introduction to this emerging field.

Classical and Discrete Differential Geometry

David Xianfeng Gu,Emil Saucan
Classical and Discrete Differential Geometry Book PDF

Author : David Xianfeng Gu,Emil Saucan
Publisher : CRC Press
Page : 690 pages
File Size : 49,9 Mb
Release : 2023-01-31
Category : Computers
ISBN : 9781000804461

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Classical and Discrete Differential Geometry by David Xianfeng Gu,Emil Saucan Pdf

This book introduces differential geometry and cutting-edge findings from the discipline by incorporating both classical approaches and modern discrete differential geometry across all facets and applications, including graphics and imaging, physics and networks. With curvature as the centerpiece, the authors present the development of differential geometry, from curves to surfaces, thence to higher dimensional manifolds; and from smooth structures to metric spaces, weighted manifolds and complexes, and to images, meshes and networks. The first part of the book is a differential geometric study of curves and surfaces in the Euclidean space, enhanced while the second part deals with higher dimensional manifolds centering on curvature by exploring the various ways of extending it to higher dimensional objects and more general structures and how to return to lower dimensional constructs. The third part focuses on computational algorithms in algebraic topology and conformal geometry, applicable for surface parameterization, shape registration and structured mesh generation. The volume will be a useful reference for students of mathematics and computer science, as well as researchers and engineering professionals who are interested in graphics and imaging, complex networks, differential geometry and curvature.

Discrete Differential Geometry

Alexander I. Bobenko,Yuri B. Suris
Discrete Differential Geometry Book PDF

Author : Alexander I. Bobenko,Yuri B. Suris
Publisher : American Mathematical Society
Page : 432 pages
File Size : 40,8 Mb
Release : 2023-09-14
Category : Mathematics
ISBN : 9781470474560

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Discrete Differential Geometry by Alexander I. Bobenko,Yuri B. Suris Pdf

An emerging field of discrete differential geometry aims at the development of discrete equivalents of notions and methods of classical differential geometry. The latter appears as a limit of a refinement of the discretization. Current interest in discrete differential geometry derives not only from its importance in pure mathematics but also from its applications in computer graphics, theoretical physics, architecture, and numerics. Rather unexpectedly, the very basic structures of discrete differential geometry turn out to be related to the theory of integrable systems. One of the main goals of this book is to reveal this integrable structure of discrete differential geometry. For a given smooth geometry one can suggest many different discretizations. Which one is the best? This book answers this question by providing fundamental discretization principles and applying them to numerous concrete problems. It turns out that intelligent theoretical discretizations are distinguished also by their good performance in applications. The intended audience of this book is threefold. It is a textbook on discrete differential geometry and integrable systems suitable for a one semester graduate course. On the other hand, it is addressed to specialists in geometry and mathematical physics. It reflects the recent progress in discrete differential geometry and contains many original results. The third group of readers at which this book is targeted is formed by specialists in geometry processing, computer graphics, architectural design, numerical simulations, and animation. They may find here answers to the question “How do we discretize differential geometry?” arising in their specific field. Prerequisites for reading this book include standard undergraduate background (calculus and linear algebra). No knowledge of differential geometry is expected, although some familiarity with curves and surfaces can be helpful.

Advances in Discrete Differential Geometry

Alexander I. Bobenko
Advances in Discrete Differential Geometry Book PDF

Author : Alexander I. Bobenko
Publisher : Springer
Page : 439 pages
File Size : 41,5 Mb
Release : 2018-04-22
Category : Mathematics
ISBN : 3662570610

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Advances in Discrete Differential Geometry by Alexander I. Bobenko Pdf

This is one of the first books on a newly emerging field of discrete differential geometry and an excellent way to access this exciting area. It surveys the fascinating connections between discrete models in differential geometry and complex analysis, integrable systems and applications in computer graphics. The authors take a closer look at discrete models in differential geometry and dynamical systems. Their curves are polygonal, surfaces are made from triangles and quadrilaterals, and time is discrete. Nevertheless, the difference between the corresponding smooth curves, surfaces and classical dynamical systems with continuous time can hardly be seen. This is the paradigm of structure-preserving discretizations. Current advances in this field are stimulated to a large extent by its relevance for computer graphics and mathematical physics. This book is written by specialists working together on a common research project. It is about differential geometry and dynamical systems, smooth and discrete theories, and on pure mathematics and its practical applications. The interaction of these facets is demonstrated by concrete examples, including discrete conformal mappings, discrete complex analysis, discrete curvatures and special surfaces, discrete integrable systems, conformal texture mappings in computer graphics, and free-form architecture. This richly illustrated book will convince readers that this new branch of mathematics is both beautiful and useful. It will appeal to graduate students and researchers in differential geometry, complex analysis, mathematical physics, numerical methods, discrete geometry, as well as computer graphics and geometry processing.

3D Printing in Mathematics

Maria Trnkova,Andrew Yarmola
3D Printing in Mathematics Book PDF

Author : Maria Trnkova,Andrew Yarmola
Publisher : American Mathematical Society
Page : 242 pages
File Size : 53,8 Mb
Release : 2023-11-07
Category : Mathematics
ISBN : 9781470469160

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3D Printing in Mathematics by Maria Trnkova,Andrew Yarmola Pdf

This volume is based on lectures delivered at the 2022 AMS Short Course “3D Printing: Challenges and Applications” held virtually from January 3–4, 2022. Access to 3D printing facilities is quickly becoming ubiquitous across college campuses. However, while equipment training is readily available, the process of taking a mathematical idea and making it into a printable model presents a big hurdle for most mathematicians. Additionally, there are still many open questions around what objects are possible to print, how to design algorithms for doing so, and what kinds of geometries have desired kinematic properties. This volume is focused on the process and applications of 3D printing for mathematical education, research, and visualization, alongside a discussion of the challenges and open mathematical problems that arise in the design and algorithmic aspects of 3D printing. The articles in this volume are focused on two main topics. The first is to make a bridge between mathematical ideas and 3D visualization. The second is to describe methods and techniques for including 3D printing in mathematical education at different levels— from pedagogy to research and from demonstrations to individual projects. We hope to establish the groundwork for engaged academic discourse on the intersections between mathematics, 3D printing and education.

Mean Field Games

François Delarue
Mean Field Games Book PDF

Author : François Delarue
Publisher : American Mathematical Society
Page : 284 pages
File Size : 55,9 Mb
Release : 2021-12-14
Category : Mathematics
ISBN : 9781470455866

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Mean Field Games by François Delarue Pdf

This volume is based on lectures delivered at the 2020 AMS Short Course “Mean Field Games: Agent Based Models to Nash Equilibria,” held January 13–14, 2020, in Denver, Colorado. Mean field game theory offers a robust methodology for studying large systems of interacting rational agents. It has been extraordinarily successful and has continued to develop since its inception. The six chapters that make up this volume provide an overview of the subject, from the foundations of the theory to applications in economics and finance, including computational aspects. The reader will find a pedagogical introduction to the main ingredients, from the forward-backward mean field game system to the master equation. Also included are two detailed chapters on the connection between finite games and mean field games, with a pedestrian description of the different methods available to solve the convergence problem. The volume concludes with two contributions on applications of mean field games and on existing numerical methods, with an opening to machine learning techniques.

An Excursion through Elementary Mathematics, Volume I

Antonio Caminha Muniz Neto
An Excursion through Elementary Mathematics Volume I Book PDF

Author : Antonio Caminha Muniz Neto
Publisher : Springer
Page : 652 pages
File Size : 54,8 Mb
Release : 2017-03-31
Category : Mathematics
ISBN : 9783319538716

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An Excursion through Elementary Mathematics, Volume I by Antonio Caminha Muniz Neto Pdf

This book provides a comprehensive, in-depth overview of elementary mathematics as explored in Mathematical Olympiads around the world. It expands on topics usually encountered in high school and could even be used as preparation for a first-semester undergraduate course. This first volume covers Real Numbers, Functions, Real Analysis, Systems of Equations, Limits and Derivatives, and much more. As part of a collection, the book differs from other publications in this field by not being a mere selection of questions or a set of tips and tricks that applies to specific problems. It starts from the most basic theoretical principles, without being either too general or too axiomatic. Examples and problems are discussed only if they are helpful as applications of the theory. Propositions are proved in detail and subsequently applied to Olympic problems or to other problems at the Olympic level. The book also explores some of the hardest problems presented at National and International Mathematics Olympiads, as well as many essential theorems related to the content. An extensive Appendix offering hints on or full solutions for all difficult problems rounds out the book.

Differential Geometry of Curves and Surfaces

Victor Andreevich Toponogov
Differential Geometry of Curves and Surfaces Book PDF

Author : Victor Andreevich Toponogov
Publisher : Springer Science & Business Media
Page : 215 pages
File Size : 55,9 Mb
Release : 2006-09-10
Category : Mathematics
ISBN : 9780817644024

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Differential Geometry of Curves and Surfaces by Victor Andreevich Toponogov Pdf

Central topics covered include curves, surfaces, geodesics, intrinsic geometry, and the Alexandrov global angle comparision theorem Many nontrivial and original problems (some with hints and solutions) Standard theoretical material is combined with more difficult theorems and complex problems, while maintaining a clear distinction between the two levels

An Excursion through Elementary Mathematics, Volume III

Antonio Caminha Muniz Neto
An Excursion through Elementary Mathematics Volume III Book PDF

Author : Antonio Caminha Muniz Neto
Publisher : Springer
Page : 648 pages
File Size : 47,5 Mb
Release : 2018-04-17
Category : Mathematics
ISBN : 9783319779775

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An Excursion through Elementary Mathematics, Volume III by Antonio Caminha Muniz Neto Pdf

This book provides a comprehensive, in-depth overview of elementary mathematics as explored in Mathematical Olympiads around the world. It expands on topics usually encountered in high school and could even be used as preparation for a first-semester undergraduate course. This third and last volume covers Counting, Generating Functions, Graph Theory, Number Theory, Complex Numbers, Polynomials, and much more. As part of a collection, the book differs from other publications in this field by not being a mere selection of questions or a set of tips and tricks that applies to specific problems. It starts from the most basic theoretical principles, without being either too general or too axiomatic. Examples and problems are discussed only if they are helpful as applications of the theory. Propositions are proved in detail and subsequently applied to Olympic problems or to other problems at the Olympic level. The book also explores some of the hardest problems presented at National and International Mathematics Olympiads, as well as many essential theorems related to the content. An extensive Appendix offering hints on or full solutions for all difficult problems rounds out the book.

A New Approach to Differential Geometry using Clifford's Geometric Algebra

John Snygg
A New Approach to Differential Geometry using Clifford s Geometric Algebra Book PDF

Author : John Snygg
Publisher : Springer Science & Business Media
Page : 476 pages
File Size : 47,6 Mb
Release : 2011-12-08
Category : Mathematics
ISBN : 9780817682828

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A New Approach to Differential Geometry using Clifford's Geometric Algebra by John Snygg Pdf

Differential geometry is the study of the curvature and calculus of curves and surfaces. A New Approach to Differential Geometry using Clifford's Geometric Algebra simplifies the discussion to an accessible level of differential geometry by introducing Clifford algebra. This presentation is relevant because Clifford algebra is an effective tool for dealing with the rotations intrinsic to the study of curved space. Complete with chapter-by-chapter exercises, an overview of general relativity, and brief biographies of historical figures, this comprehensive textbook presents a valuable introduction to differential geometry. It will serve as a useful resource for upper-level undergraduates, beginning-level graduate students, and researchers in the algebra and physics communities.

Cartan for Beginners

Thomas A. Ivey,Joseph M. Landsberg
Cartan for Beginners Book PDF

Author : Thomas A. Ivey,Joseph M. Landsberg
Publisher : American Mathematical Soc.
Page : 455 pages
File Size : 50,7 Mb
Release : 2016-12-15
Category : Exterior differential systems
ISBN : 9781470409869

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Cartan for Beginners by Thomas A. Ivey,Joseph M. Landsberg Pdf

Two central aspects of Cartan's approach to differential geometry are the theory of exterior differential systems (EDS) and the method of moving frames. This book presents thorough and modern treatments of both subjects, including their applications to both classic and contemporary problems in geometry. It begins with the classical differential geometry of surfaces and basic Riemannian geometry in the language of moving frames, along with an elementary introduction to exterior differential systems. Key concepts are developed incrementally, with motivating examples leading to definitions, theorems, and proofs. Once the basics of the methods are established, the authors develop applications and advanced topics. One notable application is to complex algebraic geometry, where they expand and update important results from projective differential geometry. As well, the book features an introduction to G-structures and a treatment of the theory of connections. The techniques of EDS are also applied to obtain explicit solutions of PDEs via Darboux's method, the method of characteristics, and Cartan's method of equivalence. This text is suitable for a one-year graduate course in differential geometry, and parts of it can be used for a one-semester course. It has numerous exercises and examples throughout. It will also be useful to experts in areas such as geometry of PDE systems and complex algebraic geometry who want to learn how moving frames and exterior differential systems apply to their fields. The second edition features three new chapters: on Riemannian geometry, emphasizing the use of representation theory; on the latest developments in the study of Darboux-integrable systems; and on conformal geometry, written in a manner to introduce readers to the related parabolic geometry perspective.

Differential Geometry

Loring W. Tu
Differential Geometry Book PDF

Author : Loring W. Tu
Publisher : Springer
Page : 347 pages
File Size : 50,6 Mb
Release : 2017-06-01
Category : Mathematics
ISBN : 9783319550848

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Differential Geometry by Loring W. Tu Pdf

This text presents a graduate-level introduction to differential geometry for mathematics and physics students. The exposition follows the historical development of the concepts of connection and curvature with the goal of explaining the Chern–Weil theory of characteristic classes on a principal bundle. Along the way we encounter some of the high points in the history of differential geometry, for example, Gauss' Theorema Egregium and the Gauss–Bonnet theorem. Exercises throughout the book test the reader’s understanding of the material and sometimes illustrate extensions of the theory. Initially, the prerequisites for the reader include a passing familiarity with manifolds. After the first chapter, it becomes necessary to understand and manipulate differential forms. A knowledge of de Rham cohomology is required for the last third of the text. Prerequisite material is contained in author's text An Introduction to Manifolds, and can be learned in one semester. For the benefit of the reader and to establish common notations, Appendix A recalls the basics of manifold theory. Additionally, in an attempt to make the exposition more self-contained, sections on algebraic constructions such as the tensor product and the exterior power are included. Differential geometry, as its name implies, is the study of geometry using differential calculus. It dates back to Newton and Leibniz in the seventeenth century, but it was not until the nineteenth century, with the work of Gauss on surfaces and Riemann on the curvature tensor, that differential geometry flourished and its modern foundation was laid. Over the past one hundred years, differential geometry has proven indispensable to an understanding of the physical world, in Einstein's general theory of relativity, in the theory of gravitation, in gauge theory, and now in string theory. Differential geometry is also useful in topology, several complex variables, algebraic geometry, complex manifolds, and dynamical systems, among other fields. The field has even found applications to group theory as in Gromov's work and to probability theory as in Diaconis's work. It is not too far-fetched to argue that differential geometry should be in every mathematician's arsenal.

Differential Geometry and Topology

Keith Burns,Marian Gidea
Differential Geometry and Topology Book PDF

Author : Keith Burns,Marian Gidea
Publisher : CRC Press
Page : 400 pages
File Size : 51,5 Mb
Release : 2005-05-27
Category : Mathematics
ISBN : 9781420057539

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Differential Geometry and Topology by Keith Burns,Marian Gidea Pdf

Accessible, concise, and self-contained, this book offers an outstanding introduction to three related subjects: differential geometry, differential topology, and dynamical systems. Topics of special interest addressed in the book include Brouwer's fixed point theorem, Morse Theory, and the geodesic flow. Smooth manifolds, Riemannian metrics, affine connections, the curvature tensor, differential forms, and integration on manifolds provide the foundation for many applications in dynamical systems and mechanics. The authors also discuss the Gauss-Bonnet theorem and its implications in non-Euclidean geometry models. The differential topology aspect of the book centers on classical, transversality theory, Sard's theorem, intersection theory, and fixed-point theorems. The construction of the de Rham cohomology builds further arguments for the strong connection between the differential structure and the topological structure. It also furnishes some of the tools necessary for a complete understanding of the Morse theory. These discussions are followed by an introduction to the theory of hyperbolic systems, with emphasis on the quintessential role of the geodesic flow. The integration of geometric theory, topological theory, and concrete applications to dynamical systems set this book apart. With clean, clear prose and effective examples, the authors' intuitive approach creates a treatment that is comprehensible to relative beginners, yet rigorous enough for those with more background and experience in the field.

Differential Geometry and Its Applications

John Oprea
Differential Geometry and Its Applications Book PDF

Author : John Oprea
Publisher : American Mathematical Soc.
Page : 469 pages
File Size : 55,9 Mb
Release : 2019-02-06
Category : Electronic
ISBN : 9781470450502

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Differential Geometry and Its Applications by John Oprea Pdf

Differential Geometry and Its Applications studies the differential geometry of surfaces with the goal of helping students make the transition from the compartmentalized courses in a standard university curriculum to a type of mathematics that is a unified whole. It mixes geometry, calculus, linear algebra, differential equations, complex variables, the calculus of variations, and notions from the sciences. That mix of ideas offers students the opportunity to visualize concepts through the use of computer algebra systems such as Maple. Differential Geometry and Its Applications emphasizes that this visualization goes hand in hand with understanding the mathematics behind the computer construction. The book is rich in results and exercises that form a continuous spectrum, from those that depend on calculation to proofs that are quite abstract.