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A D Alexandrov
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Book Synopsis A.D. Alexandrov by : S.S. Kutateladze
Download or read book A.D. Alexandrov written by S.S. Kutateladze and published by CRC Press. This book was released on 2005-07-25 with total page 442 pages. Available in PDF, EPUB and Kindle. Book excerpt: A.D. Alexandrov is considered by many to be the father of intrinsic geometry, second only to Gauss in surface theory. That appraisal stems primarily from this masterpiece--now available in its entirely for the first time since its 1948 publication in Russian. Alexandrov's treatise begins with an outline of the basic concepts, definitions, and r
Book Synopsis A. D. Alexandrov Selected Works by : Yu. G. Reshetnyak
Download or read book A. D. Alexandrov Selected Works written by Yu. G. Reshetnyak and published by CRC Press. This book was released on 2002-02-21 with total page 336 pages. Available in PDF, EPUB and Kindle. Book excerpt: Alexandr Danilovich Alexandrov has been called a giant of 20th-century mathematics. This volume contains some of the most important papers by this renowned geometer and hence, some of his most influential ideas. Alexandrov addressed a wide range of modern mathematical problems, and he did so with intelligence and elegance, solving some of the discipline's most difficult and enduring challenges. He was the first to apply many of the tools and methods of the theory of real functions and functional analysis that are now current in geometry. The topics here include convex polyhedrons and closed surfaces, an elementary proof and extension of Minkowski's theorem, Riemannian geometry and a method for Dirichlet problems. This monograph, published in English for the first time, gives unparalleled access to a brilliant mind, and advanced students and researchers in applied mathematics and geometry will find it indispensable.
Book Synopsis Convex Polyhedra by : A.D. Alexandrov
Download or read book Convex Polyhedra written by A.D. Alexandrov and published by Springer Science & Business Media. This book was released on 2005-12-08 with total page 545 pages. Available in PDF, EPUB and Kindle. Book excerpt: This classic geometry text explores the theory of 3-dimensional convex polyhedra in a unique fashion, with exceptional detail. Vital and clearly written, the book includes the basics of convex polyhedra and collects the most general existence theorems for convex polyhedra that are proved by a new and unified method. This edition includes a comprehensive bibliography by V.A. Zalgaller, and related papers as supplements to the original text.
Book Synopsis A. D. Alexandrov Selected Works Part I by : Yu. G. Reshetnyak
Download or read book A. D. Alexandrov Selected Works Part I written by Yu. G. Reshetnyak and published by CRC Press. This book was released on 2002-02-21 with total page 333 pages. Available in PDF, EPUB and Kindle. Book excerpt: Alexandr Danilovich Alexandrov has been called a giant of 20th-century mathematics. This volume contains some of the most important papers by this renowned geometer and hence, some of his most influential ideas. Alexandrov addressed a wide range of modern mathematical problems, and he did so with intelligence and elegance, solving some of the disci
Book Synopsis A.D. Alexandrov, Selected Works by : Aleksandr Danilovich Aleksandrov
Download or read book A.D. Alexandrov, Selected Works written by Aleksandr Danilovich Aleksandrov and published by . This book was released on 1996 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis A D Alexandrov by : A. D. Aleksandrov
Download or read book A D Alexandrov written by A. D. Aleksandrov and published by Gordon & Breach Science Pub. This book was released on 1998-02-01 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Alexandrov Geometry by : Stephanie Alexander
Download or read book Alexandrov Geometry written by Stephanie Alexander and published by American Mathematical Society. This book was released on 2024-05-24 with total page 303 pages. Available in PDF, EPUB and Kindle. Book excerpt: Alexandrov spaces are defined via axioms similar to those of the Euclid axioms but where certain equalities are replaced with inequalities. Depending on the signs of the inequalities, we obtain Alexandrov spaces with curvature bounded above (CBA) and curvature bounded below (CBB). Even though the definitions of the two classes of spaces are similar, their properties and known applications are quite different. The goal of this book is to give a comprehensive exposition of the structure theory of Alexandrov spaces with curvature bounded above and below. It includes all the basic material as well as selected topics inspired by considering Alexandrov spaces with CBA and with CBB simultaneously. The book also includes an extensive problem list with solutions indicated for every problem.
Book Synopsis An Invitation to Alexandrov Geometry by : Stephanie Alexander
Download or read book An Invitation to Alexandrov Geometry written by Stephanie Alexander and published by Springer. This book was released on 2019-05-08 with total page 88 pages. Available in PDF, EPUB and Kindle. Book excerpt: Aimed toward graduate students and research mathematicians, with minimal prerequisites this book provides a fresh take on Alexandrov geometry and explains the importance of CAT(0) geometry in geometric group theory. Beginning with an overview of fundamentals, definitions, and conventions, this book quickly moves forward to discuss the Reshetnyak gluing theorem and applies it to the billiards problems. The Hadamard–Cartan globalization theorem is explored and applied to construct exotic aspherical manifolds.
Book Synopsis A. D. Alexandrov Selected Works by : Yu G Reshetnyak
Download or read book A. D. Alexandrov Selected Works written by Yu G Reshetnyak and published by CRC Press. This book was released on 2019-08-30 with total page 332 pages. Available in PDF, EPUB and Kindle. Book excerpt: Alexandr Danilovich Alexandrov has been called a giant of 20th-century mathematics. This volume contains some of the most important papers by this renowned geometer and hence, some of his most influential ideas. Alexandrov addressed a wide range of modern mathematical problems, and he did so with intelligence and elegance, solving some of the discipline's most difficult and enduring challenges. He was the first to apply many of the tools and methods of the theory of real functions and functional analysis that are now current in geometry. The topics here include convex polyhedrons and closed surfaces, an elementary proof and extension of Minkowski's theorem, Riemannian geometry and a method for Dirichlet problems. This monograph, published in English for the first time, gives unparalleled access to a brilliant mind, and advanced students and researchers in applied mathematics and geometry will find it indispensable.
Book Synopsis Lectures on Spaces of Nonpositive Curvature by : Werner Ballmann
Download or read book Lectures on Spaces of Nonpositive Curvature written by Werner Ballmann and published by Birkhäuser. This book was released on 2012-12-06 with total page 114 pages. Available in PDF, EPUB and Kindle. Book excerpt: Singular spaces with upper curvature bounds and, in particular, spaces of nonpositive curvature, have been of interest in many fields, including geometric (and combinatorial) group theory, topology, dynamical systems and probability theory. In the first two chapters of the book, a concise introduction into these spaces is given, culminating in the Hadamard-Cartan theorem and the discussion of the ideal boundary at infinity for simply connected complete spaces of nonpositive curvature. In the third chapter, qualitative properties of the geodesic flow on geodesically complete spaces of nonpositive curvature are discussed, as are random walks on groups of isometries of nonpositively curved spaces. The main class of spaces considered should be precisely complementary to symmetric spaces of higher rank and Euclidean buildings of dimension at least two (Rank Rigidity conjecture). In the smooth case, this is known and is the content of the Rank Rigidity theorem. An updated version of the proof of the latter theorem (in the smooth case) is presented in Chapter IV of the book. This chapter contains also a short introduction into the geometry of the unit tangent bundle of a Riemannian manifold and the basic facts about the geodesic flow. In an appendix by Misha Brin, a self-contained and short proof of the ergodicity of the geodesic flow of a compact Riemannian manifold of negative curvature is given. The proof is elementary and should be accessible to the non-specialist. Some of the essential features and problems of the ergodic theory of smooth dynamical systems are discussed, and the appendix can serve as an introduction into this theory.
Book Synopsis A.D. Alexandrov, Selected Works: Intrinsic geometry of convex surfaces by : Aleksandr Danilovich Aleksandrov
Download or read book A.D. Alexandrov, Selected Works: Intrinsic geometry of convex surfaces written by Aleksandr Danilovich Aleksandrov and published by . This book was released on 1996 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis A.D. Alexandrov by : S.S. Kutateladze
Download or read book A.D. Alexandrov written by S.S. Kutateladze and published by Chapman and Hall/CRC. This book was released on 2005-07-25 with total page 440 pages. Available in PDF, EPUB and Kindle. Book excerpt: A.D. Alexandrov is considered by many to be the father of intrinsic geometry, second only to Gauss in surface theory. That appraisal stems primarily from this masterpiece--now available in its entirely for the first time since its 1948 publication in Russian. Alexandrov's treatise begins with an outline of the basic concepts, definitions, and results relevant to intrinsic geometry. It reviews the general theory, then presents the requisite general theorems on rectifiable curves and curves of minimum length. Proof of some of the general properties of the intrinsic metric of convex surfaces follows. The study then splits into two almost independent lines: further exploration of the intrinsic geometry of convex surfaces and proof of the existence of a surface with a given metric. The final chapter reviews the generalization of the whole theory to convex surfaces in the Lobachevskii space and in the spherical space, concluding with an outline of the theory of nonconvex surfaces. Alexandrov's work was both original and extremely influential. This book gave rise to studying surfaces "in the large," rejecting the limitations of smoothness, and reviving the style of Euclid. Progress in geometry in recent decades correlates with the resurrection of the synthetic methods of geometry and brings the ideas of Alexandrov once again into focus. This text is a classic that remains unsurpassed in its clarity and scope.
Download or read book Perfect Rigour written by Masha Gessen and published by Icon Books Ltd. This book was released on 2011-03-03 with total page 119 pages. Available in PDF, EPUB and Kindle. Book excerpt: In 2006, an eccentric Russian mathematician named Grigori Perelman solved one of the world's greatest intellectual puzzles. The Poincare conjecture is an extremely complex topological problem that had eluded the best minds for over a century. In 2000, the Clay Institute in Boston named it one of seven great unsolved mathematical problems, and promised a million dollars to anyone who could find a solution. Perelman was awarded the prize this year - and declined the money. Journalist Masha Gessen was determined to find out why. Drawing on interviews with Perelman's teachers, classmates, coaches, teammates, and colleagues in Russia and the US - and informed by her own background as a math whiz raised in Russia - she set out to uncover the nature of Perelman's astonishing abilities. In telling his story, Masha Gessen has constructed a gripping and tragic tale that sheds rare light on the unique burden of genius.
Book Synopsis Nonlinear Elliptic Equations and Nonassociative Algebras by : Nikolai Nadirashvili
Download or read book Nonlinear Elliptic Equations and Nonassociative Algebras written by Nikolai Nadirashvili and published by American Mathematical Soc.. This book was released on 2014-12-03 with total page 250 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents applications of noncommutative and nonassociative algebras to constructing unusual (nonclassical and singular) solutions to fully nonlinear elliptic partial differential equations of second order. The methods described in the book are used to solve a longstanding problem of the existence of truly weak, nonsmooth viscosity solutions. Moreover, the authors provide an almost complete description of homogeneous solutions to fully nonlinear elliptic equations. It is shown that even in the very restricted setting of "Hessian equations", depending only on the eigenvalues of the Hessian, these equations admit homogeneous solutions of all orders compatible with known regularity for viscosity solutions provided the space dimension is five or larger. To the contrary, in dimension four or less the situation is completely different, and our results suggest strongly that there are no nonclassical homogeneous solutions at all in dimensions three and four. Thus this book gives a complete list of dimensions where nonclassical homogeneous solutions to fully nonlinear uniformly elliptic equations do exist; this should be compared with the situation of, say, ten years ago when the very existence of nonclassical viscosity solutions was not known.
Book Synopsis Reshetnyak's Theory of Subharmonic Metrics by : François Fillastre
Download or read book Reshetnyak's Theory of Subharmonic Metrics written by François Fillastre and published by Springer Nature. This book was released on 2023-10-20 with total page 389 pages. Available in PDF, EPUB and Kindle. Book excerpt: Despite the fundamental role played by Reshetnyak's work in the theory of surfaces of bounded integral curvature, the proofs of his results were only available in his original articles, written in Russian and often hard to find. This situation used to be a serious problem for experts in the field. This book provides English translations of the full set of Reshetnyak's articles on the subject. Together with the companion articles, this book provides an accessible and comprehensive reference for the subject. In turn, this book should concern any researcher (confirmed or not) interested in, or active in, the field of bounded integral curvature surfaces, or more generally interested in surface geometry and geometric analysis. Due to the analytic nature of Reshetnyak's approach, it appears that his articles are very accessible for a modern audience, comparing to the works using a more synthetic approach. These articles of Reshetnyak concern more precisely the work carried by the author following the completion of his PhD thesis, under the supervision of A.D. Alexandrov. Over the period from the 1940’s to the 1960’s, the Leningrad School of Geometry, developed a theory of the metric geometry of surfaces, similar to the classical theory of Riemannian surfaces, but with lower regularity, allowing greater flexibility. Let us mention A.D. Alexandrov, Y.D. Burago and V.A. Zalgaller. The types of surfaces studied by this school are now known as surfaces of bounded curvature. Particular cases are that of surfaces with curvature bounded from above or below, the study of which gained special attention after the works of M. Gromov and G. Perelman. Nowadays, these concepts have been generalized to higher dimensions, to graphs, and so on, and the study of metrics of weak regularity remains an active and challenging field. Reshetnyak developed an alternative and analytic approach to surfaces of bounded integral curvature. The underlying idea is based on the theorem of Gauss which states that every Riemannian surface is locally conformal to Euclidean space. Reshetnyak thus studied generalized metrics which are locally conformal to the Euclidean metric with conformal factor given by the logarithm of the difference between two subharmonic functions on the plane. Reshetnyak's condition appears to provide the correct regularity required to generalize classical concepts such as measure of curvature, integral geodesic curvature for curves, and so on, and in turn, to recover surfaces of bounded curvature. Chapter-No.7, Chapter-No.8, Chapter-No.12 and Chapter-No.13 are available open access under Creative Commons Attribution-NonCommercial 4.0 International License via link.springer.com.
Book Synopsis Metric Spaces of Non-Positive Curvature by : Martin R. Bridson
Download or read book Metric Spaces of Non-Positive Curvature written by Martin R. Bridson and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 665 pages. Available in PDF, EPUB and Kindle. Book excerpt: A description of the global properties of simply-connected spaces that are non-positively curved in the sense of A. D. Alexandrov, and the structure of groups which act on such spaces by isometries. The theory of these objects is developed in a manner accessible to anyone familiar with the rudiments of topology and group theory: non-trivial theorems are proved by concatenating elementary geometric arguments, and many examples are given. Part I provides an introduction to the geometry of geodesic spaces, while Part II develops the basic theory of spaces with upper curvature bounds. More specialized topics, such as complexes of groups, are covered in Part III.
Book Synopsis Metric Spaces, Convexity and Nonpositive Curvature by : Athanase Papadopoulos
Download or read book Metric Spaces, Convexity and Nonpositive Curvature written by Athanase Papadopoulos and published by European Mathematical Society. This book was released on 2005 with total page 306 pages. Available in PDF, EPUB and Kindle. Book excerpt:
