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Invariant Algebras And Geometric Reasoning
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Book Synopsis Invariant Algebras and Geometric Reasoning by : Hongbo Li
Download or read book Invariant Algebras and Geometric Reasoning written by Hongbo Li and published by World Scientific. This book was released on 2008 with total page 533 pages. Available in PDF, EPUB and Kindle. Book excerpt: The demand for more reliable geometric computing in robotics, computer vision and graphics has revitalized many venerable algebraic subjects in mathematics OCo among them, GrassmannOCoCayley algebra and Geometric Algebra. Nowadays, they are used as powerful languages for projective, Euclidean and other classical geometries. This book contains the author and his collaborators' most recent, original development of GrassmannOCoCayley algebra and Geometric Algebra and their applications in automated reasoning of classical geometries. It includes two of the three advanced invariant algebras OCo Cayley bracket algebra, conformal geometric algebra, and null bracket algebra OCo for highly efficient geometric computing. They form the theory of advanced invariants, and capture the intrinsic beauty of geometric languages and geometric computing. Apart from their applications in discrete and computational geometry, the new languages are currently being used in computer vision, graphics and robotics by many researchers worldwide. Sample Chapter(s). Chapter 1: Introduction (252 KB). Contents: Projective Space, Bracket Algebra and GrassmannOCoCayley Algebra; Projective Incidence Geometry with Cayley Bracket Algebra; Projective Conic Geometry with Bracket Algebra and Quadratic Grassmann-Cayley Algebra; Inner-product Bracket Algebra and Clifford Algebra; Geometric Algebra; Euclidean Geometry and Conformal GrassmannOCoCayley Algebra; Conformal Clifford Algebra and Classical Geometries. Readership: Graduate students in discrete and computational geometry, and computer mathematics; mathematicians and computer scientists.
Book Synopsis Invariant Algebras and Geometric Reasoning by : Hongbo Li
Download or read book Invariant Algebras and Geometric Reasoning written by Hongbo Li and published by World Scientific. This book was released on 2008 with total page 533 pages. Available in PDF, EPUB and Kindle. Book excerpt: A moving portrait of Africa from Polands most celebrated foreign correspondent - a masterpiece from a modern master. Famous for being in the wrong places at just the right times, Ryszard Kapuscinski arrived in Africa in 1957, at the beginning of the end of colonial rule - the &"sometimes dramatic and painful, sometimes enjoyable and jubilant&" rebirth of a continent.The Shadow of the Sunsums up the authors experiences (&"the record of a 40-year marriage&") in this place that became the central obsession of his remarkable career. From the hopeful years of independence through the bloody disintegration of places like Nigeria, Rwanda and Angola, Kapuscinski recounts great social and political changes through the prism of the ordinary African. He examines the rough-and-ready physical world and identifies the true geography of Africa: a little-understood spiritual universe, an African way of being. He looks also at Africa in the wake of two epoch-making changes: the arrival of AIDS and the definitive departure of the white man. Kapuscinskis rare humanity invests his subjects with a grandeur and a dignity unmatched by any other writer on the Third World, and his unique ability to discern the universal in the particular has never been more powerfully displayed than in this work. From the Trade Paperback edition.
Book Synopsis Actions and Invariants of Algebraic Groups by : Walter Ricardo Ferrer Santos
Download or read book Actions and Invariants of Algebraic Groups written by Walter Ricardo Ferrer Santos and published by CRC Press. This book was released on 2017-09-19 with total page 542 pages. Available in PDF, EPUB and Kindle. Book excerpt: Actions and Invariants of Algebraic Groups, Second Edition presents a self-contained introduction to geometric invariant theory starting from the basic theory of affine algebraic groups and proceeding towards more sophisticated dimensions." Building on the first edition, this book provides an introduction to the theory by equipping the reader with the tools needed to read advanced research in the field. Beginning with commutative algebra, algebraic geometry and the theory of Lie algebras, the book develops the necessary background of affine algebraic groups over an algebraically closed field, and then moves toward the algebraic and geometric aspects of modern invariant theory and quotients.
Book Synopsis Mathematical Aspects of Computer and Information Sciences by : Ilias S. Kotsireas
Download or read book Mathematical Aspects of Computer and Information Sciences written by Ilias S. Kotsireas and published by Springer. This book was released on 2016-04-16 with total page 628 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the thoroughly refereed post-conference proceedings of the 6th International Conference on Mathematical Aspects of Computer and Information Sciences, MACIS 2015, held in Berlin, Germany, in November 2015. The 48 revised papers presented together with 7 invited papers were carefully reviewed and selected from numerous submissions. The papers are grouped in topical sections on curves and surfaces, applied algebraic geometry, cryptography, verified numerical computation, polynomial system solving, managing massive data, computational theory of differential and difference equations, data and knowledge exploration, algorithm engineering in geometric computing, real complexity: theory and practice, global optimization, and general session.
Book Synopsis Invariant Methods in Discrete and Computational Geometry by : Neil L. White
Download or read book Invariant Methods in Discrete and Computational Geometry written by Neil L. White and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 331 pages. Available in PDF, EPUB and Kindle. Book excerpt: Invariant, or coordinate-free methods provide a natural framework for many geometric questions. Invariant Methods in Discrete and Computational Geometry provides a basic introduction to several aspects of invariant theory, including the supersymmetric algebra, the Grassmann-Cayler algebra, and Chow forms. It also presents a number of current research papers on invariant theory and its applications to problems in geometry, such as automated theorem proving and computer vision. Audience: Researchers studying mathematics, computers and robotics.
Book Synopsis Guide to Geometric Algebra in Practice by : Leo Dorst
Download or read book Guide to Geometric Algebra in Practice written by Leo Dorst and published by Springer Science & Business Media. This book was released on 2011-08-28 with total page 458 pages. Available in PDF, EPUB and Kindle. Book excerpt: This highly practical Guide to Geometric Algebra in Practice reviews algebraic techniques for geometrical problems in computer science and engineering, and the relationships between them. The topics covered range from powerful new theoretical developments, to successful applications, and the development of new software and hardware tools. Topics and features: provides hands-on review exercises throughout the book, together with helpful chapter summaries; presents a concise introductory tutorial to conformal geometric algebra (CGA) in the appendices; examines the application of CGA for the description of rigid body motion, interpolation and tracking, and image processing; reviews the employment of GA in theorem proving and combinatorics; discusses the geometric algebra of lines, lower-dimensional algebras, and other alternatives to 5-dimensional CGA; proposes applications of coordinate-free methods of GA for differential geometry.
Book Synopsis Geometric Invariant Theory, Holomorphic Vector Bundles and the Harder-Narasimhan Filtration by : Alfonso Zamora Saiz
Download or read book Geometric Invariant Theory, Holomorphic Vector Bundles and the Harder-Narasimhan Filtration written by Alfonso Zamora Saiz and published by . This book was released on 2021 with total page 127 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book introduces key topics on Geometric Invariant Theory, a technique to obtaining quotients in algebraic geometry with a good set of properties, through various examples. It starts from the classical Hilbert classification of binary forms, advancing to the construction of the moduli space of semistable holomorphic vector bundles, and to Hitchin's theory on Higgs bundles. The relationship between the notion of stability between algebraic, differential and symplectic geometry settings is also covered. Unstable objects in moduli problems -- a result of the construction of moduli spaces -- get specific attention in this work. The notion of the Harder-Narasimhan filtration as a tool to handle them, and its relationship with GIT quotients, provide instigating new calculations in several problems. Applications include a survey of research results on correspondences between Harder-Narasimhan filtrations with the GIT picture and stratifications of the moduli space of Higgs bundles. Graduate students and researchers who want to approach Geometric Invariant Theory in moduli constructions can greatly benefit from this reading, whose key prerequisites are general courses on algebraic geometry and differential geometry.
Book Synopsis Lectures on Invariant Theory by : Igor Dolgachev
Download or read book Lectures on Invariant Theory written by Igor Dolgachev and published by Cambridge University Press. This book was released on 2003-08-07 with total page 244 pages. Available in PDF, EPUB and Kindle. Book excerpt: The primary goal of this 2003 book is to give a brief introduction to the main ideas of algebraic and geometric invariant theory. It assumes only a minimal background in algebraic geometry, algebra and representation theory. Topics covered include the symbolic method for computation of invariants on the space of homogeneous forms, the problem of finite-generatedness of the algebra of invariants, the theory of covariants and constructions of categorical and geometric quotients. Throughout, the emphasis is on concrete examples which originate in classical algebraic geometry. Based on lectures given at University of Michigan, Harvard University and Seoul National University, the book is written in an accessible style and contains many examples and exercises. A novel feature of the book is a discussion of possible linearizations of actions and the variation of quotients under the change of linearization. Also includes the construction of toric varieties as torus quotients of affine spaces.
Book Synopsis Lie Groups and Geometric Aspects of Isometric Actions by : Marcos M. Alexandrino
Download or read book Lie Groups and Geometric Aspects of Isometric Actions written by Marcos M. Alexandrino and published by Springer. This book was released on 2015-05-22 with total page 213 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides quick access to the theory of Lie groups and isometric actions on smooth manifolds, using a concise geometric approach. After a gentle introduction to the subject, some of its recent applications to active research areas are explored, keeping a constant connection with the basic material. The topics discussed include polar actions, singular Riemannian foliations, cohomogeneity one actions, and positively curved manifolds with many symmetries. This book stems from the experience gathered by the authors in several lectures along the years and was designed to be as self-contained as possible. It is intended for advanced undergraduates, graduate students and young researchers in geometry and can be used for a one-semester course or independent study.
Book Synopsis Algebraic Geometry IV by : A.N. Parshin
Download or read book Algebraic Geometry IV written by A.N. Parshin and published by Springer Science & Business Media. This book was released on 1994-04-25 with total page 304 pages. Available in PDF, EPUB and Kindle. Book excerpt: Two contributions on closely related subjects: the theory of linear algebraic groups and invariant theory, by well-known experts in the fields. The book will be very useful as a reference and research guide to graduate students and researchers in mathematics and theoretical physics.
Book Synopsis Real Spinorial Groups by : Sebastià Xambó-Descamps
Download or read book Real Spinorial Groups written by Sebastià Xambó-Descamps and published by Springer. This book was released on 2018-11-22 with total page 151 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book explores the Lipschitz spinorial groups (versor, pinor, spinor and rotor groups) of a real non-degenerate orthogonal geometry (or orthogonal geometry, for short) and how they relate to the group of isometries of that geometry. After a concise mathematical introduction, it offers an axiomatic presentation of the geometric algebra of an orthogonal geometry. Once it has established the language of geometric algebra (linear grading of the algebra; geometric, exterior and interior products; involutions), it defines the spinorial groups, demonstrates their relation to the isometry groups, and illustrates their suppleness (geometric covariance) with a variety of examples. Lastly, the book provides pointers to major applications, an extensive bibliography and an alphabetic index. Combining the characteristics of a self-contained research monograph and a state-of-the-art survey, this book is a valuable foundation reference resource on applications for both undergraduate and graduate students.
Book Synopsis Algebraic Invariants by : Leonard Eugene Dickson
Download or read book Algebraic Invariants written by Leonard Eugene Dickson and published by CreateSpace. This book was released on 2014-02-11 with total page 110 pages. Available in PDF, EPUB and Kindle. Book excerpt: AUTHOR'S PREFACE. This introduction to the classical theory of invariants of algebraic forms is divided into three parts of approximately equal length. Part I treats of linear transformations both from the standpoint of a change of the two points of reference or the triangle of reference used in the definition of the homogeneous coordinates of points in a line or plane, and also from the standpoint of projective geometry. Examples are given of invariants of forms f/ of low degrees in two or three variables, and the vanishing of an invariant of f is shown to give a geometrical property of the locus f = 0, which, on the one hand, is independent of the points of reference or triangle of reference, and, on the other hand, is unchanged by projection. Certain covariants such as Jacobians and Hessians are discussed and their algebraic and geometrical interpretations given; in particular, the use of the Hessian in the solution of a cubic equation and in the discussion of the points of inflexion of a plane cubic curve. In brief, beginning with ample illustrations from plane analytics, the reader is led by easy stages to the standpoint of linear transformations, their invariants and interpretations, employed in analytic projective geometry and modern algebra. Part II treats of the algebraic properties of invariants and covariants, chiefly of binary forms: homogeneity, weight, annihilators, semi-invariant leaders of covariants, law of reciprocity, fundamental systems, properties as functions of the roots, and production by means of differential operators. Any quartic equation is solved by reducing it to a canonical form by means of the Hessian (§ 33). Irrational invariants are illustrated by a carefully selected set of exercises (§ 35). Part III gives an introduction to the symbolic notation of Aronhold and Clebsch. The notation is first explained at length for a simple case; likewise the fundamental theorem on the types of symbolic factors of a term of a covariant of binary forms is first proved for a simple example by the method later used for the general theorem. In view of these and similar attentions to the needs of those making their first acquaintance with the symbolic notation, the difficulties usually encountered will, it is believed, be largely avoided. This notation must be mastered by those who would go deeply into the theory of invariants and its applications. Hilbert's theorem on the expression of the forms of a set linearly in terms of a finite number of forms of the set is proved and applied to establish the finiteness of a fundamental set of covariants of a system of binary forms. The theory of transvectants is developed as far as needed in the discussion of apolarity of binary forms and its application to rational curves (§§ 53-57), and in the determination by induction of a fundamental system of covariants of a binary form without the aid of the more technical supplementary concepts employed by Gordan. Finally, there is a discussion of the types of symbolic factors in any term of a concomitant of a system of forms in three or four variables, with remarks on fine and plane coordinates. For further developments reference is made at appropriate places to the texts in English by Salmon, Elliott, and Grace and Young, as well as to Gordan's Invariantentheorie. The standard work on the geometrical side of invariants is Clebsch-Lindemann, Vorlesungen über Geometrie. Reference may be made to books by W. F. Meyer, Apolaritdt und Rationale Curve, Bericht uber den gegenwarligen Stand der Invariantentheorie, and Formentheorie. Concerning invariant-factors, elementary divisors, and pairs of quadratic or bilinear forms, not treated here, see Muth, Elementartheiler, Bromwich, Quadratic Forms and their Classification by Means of Invariant Factors, and Bocher's Introduction to Higher Algebra....
Book Synopsis Lectures on the Geometry of Manifolds by : L I Nicolaescu
Download or read book Lectures on the Geometry of Manifolds written by L I Nicolaescu and published by World Scientific. This book was released on 1996-11-13 with total page 500 pages. Available in PDF, EPUB and Kindle. Book excerpt: The object of this book is to introduce the reader to some of the most important techniques of modern global geometry. In writing it we had in mind the beginning graduate student willing to specialize in this very challenging field of mathematics. The necessary prerequisite is a good knowledge of the calculus with several variables, linear algebra and some elementary point-set topology. We tried to address several issues. 1. The Language; 2. The Problems; 3. The Methods; 4. The Answers. Historically, the problems came first, then came the methods and the language while the answers came last. The space constraints forced us to change this order and we had to painfully restrict our selection of topics to be covered. This process always involves a loss of intuition and we tried to balance this by offering as many examples and pictures as often as possible. We test most of our results and techniques on two basic classes examples: surfaces (which can be easily visualized) and Lie groups (which can be elegantly algebraized). When possible we present several facets of the same issue. We believe that a good familiarity with the formalism of differential geometry is absolutely necessary in understanding and solving concrete problems and this is why we presented it in some detail. Every new concept is supported by concrete examples interesting not only from an academic point of view. Our interest is mainly in global questions and in particular the interdependence geometry ↔ topology, local ↔ global. We had to develop many algebraico-topological techniques in the special context of smooth manifolds. We spent a big portion of this book discussing the DeRham cohomology and its ramifications: Poincaré duality, intersection theory, degree theory, Thom isomorphism, characteristic classes, Gauss–Bonnet etc. We tried to calculate the cohomology groups of as many as possible concrete examples and we had to do this without relying on the powerful apparatus of homotopy theory (CW-complexes etc.). Some of the proofs are not the most direct ones but the means are sometimes more interesting than the ends. For example in computing the cohomology of complex grassmannians we returned to classical invariant theory and used some brilliant but unadvertised old ideas. In the last part of the book we discuss elliptic partial differential equations. This requires a familiarity with functional analysis. We painstakingly described the proofs of elliptic Lp and Hölder estimates (assuming some deep results of harmonic analysis) for arbitrary elliptic operators with smooth coefficients. It is not a “light meal” but the ideas are useful in a large number of instances. We present a few applications of these techniques (Hodge theory, uniformization theorem). We conclude with a close look to a very important class of elliptic operators namely the Dirac operators. We discuss their algebraic structure in some detail, Weizenböck formulæ and many concrete examples. Contents:ManifoldsNatural Constructions on ManifoldsCalculus on ManifoldsRiemannian GeometryElements of the Calculus of VariationsThe Fundamental Group and Covering SpacesCohomologyCharacteristic ClassesElliptic Equations on ManifoldsDirac OperatorsBibliographyIndex Readership: Mathematicians. keywords:Calculus on Manifolds;Riemannian Geometry;Vector Bundles and Connections;DeRham Cohomology;Characteristic Classes;Elliptic Partial Differential Equations on Manifolds;Hodge Theory;Dirac Operators “… the greatest virtue of the book, is its presentation of a large number of interesting, significant, and up-to-the-minute examples … In all, this would be an excellent text and reference for an introductory course or series of courses in this active area of mathematics.” Mathematical Reviews “The book is marked by its clear presentation, contains many exercises and is illustrated by numerous detailed examples.” Mathematics Abstracts
Book Synopsis Enumerative Invariants in Algebraic Geometry and String Theory by : Marcos Marino
Download or read book Enumerative Invariants in Algebraic Geometry and String Theory written by Marcos Marino and published by Springer Science & Business Media. This book was released on 2008-08-22 with total page 219 pages. Available in PDF, EPUB and Kindle. Book excerpt: Starting in the middle of the 80s, there has been a growing and fruitful interaction between algebraic geometry and certain areas of theoretical high-energy physics, especially the various versions of string theory. Physical heuristics have provided inspiration for new mathematical definitions (such as that of Gromov-Witten invariants) leading in turn to the solution of problems in enumerative geometry. Conversely, the availability of mathematically rigorous definitions and theorems has benefited the physics research by providing the required evidence in fields where experimental testing seems problematic. The aim of this volume, a result of the CIME Summer School held in Cetraro, Italy, in 2005, is to cover part of the most recent and interesting findings in this subject.
Book Synopsis An Introduction to Lie Groups and the Geometry of Homogeneous Spaces by : Andreas Arvanitogeōrgos
Download or read book An Introduction to Lie Groups and the Geometry of Homogeneous Spaces written by Andreas Arvanitogeōrgos and published by American Mathematical Soc.. This book was released on 2003 with total page 162 pages. Available in PDF, EPUB and Kindle. Book excerpt: It is remarkable that so much about Lie groups could be packed into this small book. But after reading it, students will be well-prepared to continue with more advanced, graduate-level topics in differential geometry or the theory of Lie groups. The theory of Lie groups involves many areas of mathematics. In this book, Arvanitoyeorgos outlines enough of the prerequisites to get the reader started. He then chooses a path through this rich and diverse theory that aims for an understanding of the geometry of Lie groups and homogeneous spaces. In this way, he avoids the extra detail needed for a thorough discussion of other topics. Lie groups and homogeneous spaces are especially useful to study in geometry, as they provide excellent examples where quantities (such as curvature) are easier to compute. A good understanding of them provides lasting intuition, especially in differential geometry. The book is suitable for advanced undergraduates, graduate students, and research mathematicians interested in differential geometry and neighboring fields, such as topology, harmonic analysis, and mathematical physics.
Book Synopsis Algebraic Groups and Their Birational Invariants by : Valentin Evgenʹevich Voskresenskiĭ
Download or read book Algebraic Groups and Their Birational Invariants written by Valentin Evgenʹevich Voskresenskiĭ and published by American Mathematical Soc.. This book was released on 1998 with total page 240 pages. Available in PDF, EPUB and Kindle. Book excerpt: Since the late 1960s, methods of birational geometry have been used successfully in the theory of linear algebraic groups, especially in arithmetic problems. This book studies birational properties of linear algebraic groups focusing on arithmetic applications.
Book Synopsis Algebraic Homogeneous Spaces and Invariant Theory by : Frank D. Grosshans
Download or read book Algebraic Homogeneous Spaces and Invariant Theory written by Frank D. Grosshans and published by . This book was released on 2014-09-01 with total page 160 pages. Available in PDF, EPUB and Kindle. Book excerpt:
