Download or read book Geometry in Action written by Clark Kimberling and published by Springer Science & Business Media. This book was released on 2003 with total page 212 pages. Available in PDF, EPUB and Kindle. Book excerpt: Geometry in Action uses Sketchpad? to awaken student creativity through discovery-based learning. It supplements any college geometry course in which The Geometer's Sketchpad is used. All students must have access to The Geometer's Sketchpad.Each book is packaged with a CD-ROM for students that illustrates what is meant by geometry in action. Students explore 27 sketches prepared by the author to demonstrate Sketchpad's capabilities by dragging points to see shifts in graphs, by animating tesselations to create new patterns, and much, much more! Also included on this CD is the Poincare Disk, a Sketchpad file used to dig deeper into non-Euclidean geometry with The Geometer's Sketchpad.
Download or read book From Groups to Geometry and Back written by Vaughn Climenhaga and published by American Mathematical Soc.. This book was released on 2017-04-07 with total page 442 pages. Available in PDF, EPUB and Kindle. Book excerpt: Groups arise naturally as symmetries of geometric objects, and so groups can be used to understand geometry and topology. Conversely, one can study abstract groups by using geometric techniques and ultimately by treating groups themselves as geometric objects. This book explores these connections between group theory and geometry, introducing some of the main ideas of transformation groups, algebraic topology, and geometric group theory. The first half of the book introduces basic notions of group theory and studies symmetry groups in various geometries, including Euclidean, projective, and hyperbolic. The classification of Euclidean isometries leads to results on regular polyhedra and polytopes; the study of symmetry groups using matrices leads to Lie groups and Lie algebras. The second half of the book explores ideas from algebraic topology and geometric group theory. The fundamental group appears as yet another group associated to a geometric object and turns out to be a symmetry group using covering spaces and deck transformations. In the other direction, Cayley graphs, planar models, and fundamental domains appear as geometric objects associated to groups. The final chapter discusses groups themselves as geometric objects, including a gentle introduction to Gromov's theorem on polynomial growth and Grigorchuk's example of intermediate growth. The book is accessible to undergraduate students (and anyone else) with a background in calculus, linear algebra, and basic real analysis, including topological notions of convergence and connectedness. This book is a result of the MASS course in algebra at Penn State University in the fall semester of 2009.
Download or read book Geometry Rigidity and Group Actions written by Robert J. Zimmer and published by University of Chicago Press. This book was released on 2011-04-15 with total page 659 pages. Available in PDF, EPUB and Kindle. Book excerpt: The study of group actions is more than 100 years old but remains a widely studied topic in a variety of mathematic fields. A central development in the last 50 years is the phenomenon of rigidity, whereby one can classify actions of certain groups. This book looks at rigidity.
Download or read book Geometry Mechanics and Control in Action for the Falling Cat written by Toshihiro Iwai and published by Springer Nature. This book was released on 2021-04-23 with total page 193 pages. Available in PDF, EPUB and Kindle. Book excerpt: The falling cat is an interesting theme to pursue, in which geometry, mechanics, and control are in action together. As is well known, cats can almost always land on their feet when tossed into the air in an upside-down attitude. If cats are not given a non-vanishing angular momentum at an initial instant, they cannot rotate during their motion, and the motion they can make in the air is vibration only. However, cats accomplish a half turn without rotation when landing on their feet. In order to solve this apparent mystery, one needs to thoroughly understand rotations and vibrations. The connection theory in differential geometry can provide rigorous definitions of rotation and vibration for many-body systems. Deformable bodies of cats are not easy to treat mechanically. A feasible way to approach the question of the falling cat is to start with many-body systems and then proceed to rigid bodies and, further, to jointed rigid bodies, which can approximate the body of a cat. In this book, the connection theory is applied first to a many-body system to show that vibrational motions of the many-body system can result in rotations without performing rotational motions and then to the cat model consisting of jointed rigid bodies. On the basis of this geometric setting, mechanics of many-body systems and of jointed rigid bodies must be set up. In order to take into account the fact that cats can deform their bodies, three torque inputs which may give a twist to the cat model are applied as control inputs under the condition of the vanishing angular momentum. Then, a control is designed according to the port-controlled Hamiltonian method for the model cat to perform a half turn and to halt the motion upon landing. The book also gives a brief review of control systems through simple examples to explain the role of control inputs.
Download or read book The Principle of Least Action in Geometry and Dynamics written by Karl Friedrich Siburg and published by Springer Science & Business Media. This book was released on 2004-05-17 with total page 148 pages. Available in PDF, EPUB and Kindle. Book excerpt: New variational methods by Aubry, Mather, and Mane, discovered in the last twenty years, gave deep insight into the dynamics of convex Lagrangian systems. This book shows how this Principle of Least Action appears in a variety of settings (billiards, length spectrum, Hofer geometry, modern symplectic geometry). Thus, topics from modern dynamical systems and modern symplectic geometry are linked in a new and sometimes surprising way. The central object is Mather’s minimal action functional. The level is for graduate students onwards, but also for researchers in any of the subjects touched in the book.
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 215 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.
Download or read book Spectral Action in Noncommutative Geometry written by Michał Eckstein and published by Springer. This book was released on 2018-12-18 with total page 155 pages. Available in PDF, EPUB and Kindle. Book excerpt: What is spectral action, how to compute it and what are the known examples? This book offers a guided tour through the mathematical habitat of noncommutative geometry à la Connes, deliberately unveiling the answers to these questions. After a brief preface flashing the panorama of the spectral approach, a concise primer on spectral triples is given. Chapter 2 is designed to serve as a toolkit for computations. The third chapter offers an in-depth view into the subtle links between the asymptotic expansions of traces of heat operators and meromorphic extensions of the associated spectral zeta functions. Chapter 4 studies the behaviour of the spectral action under fluctuations by gauge potentials. A subjective list of open problems in the field is spelled out in the fifth Chapter. The book concludes with an appendix including some auxiliary tools from geometry and analysis, along with examples of spectral geometries. The book serves both as a compendium for researchers in the domain of noncommutative geometry and an invitation to mathematical physicists looking for new concepts.
Download or read book Computational Geometry written by Mark de Berg and published by Springer Science & Business Media. This book was released on 2013-04-17 with total page 370 pages. Available in PDF, EPUB and Kindle. Book excerpt: This introduction to computational geometry focuses on algorithms. Motivation is provided from the application areas as all techniques are related to particular applications in robotics, graphics, CAD/CAM, and geographic information systems. Modern insights in computational geometry are used to provide solutions that are both efficient and easy to understand and implement.
Download or read book The Geometry of Discrete Groups written by Alan F. Beardon and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 350 pages. Available in PDF, EPUB and Kindle. Book excerpt: This text is intended to serve as an introduction to the geometry of the action of discrete groups of Mobius transformations. The subject matter has now been studied with changing points of emphasis for over a hundred years, the most recent developments being connected with the theory of 3-manifolds: see, for example, the papers of Poincare [77] and Thurston [101]. About 1940, the now well-known (but virtually unobtainable) Fenchel-Nielsen manuscript appeared. Sadly, the manuscript never appeared in print, and this more modest text attempts to display at least some of the beautiful geo metrical ideas to be found in that manuscript, as well as some more recent material. The text has been written with the conviction that geometrical explana tions are essential for a full understanding of the material and that however simple a matrix proof might seem, a geometric proof is almost certainly more profitable. Further, wherever possible, results should be stated in a form that is invariant under conjugation, thus making the intrinsic nature of the result more apparent. Despite the fact that the subject matter is concerned with groups of isometries of hyperbolic geometry, many publications rely on Euclidean estimates and geometry. However, the recent developments have again emphasized the need for hyperbolic geometry, and I have included a comprehensive chapter on analytical (not axiomatic) hyperbolic geometry. It is hoped that this chapter will serve as a "dictionary" offormulae in plane hyperbolic geometry and as such will be of interest and use in its own right.
Download or read book Geometry and Dynamics of Groups and Spaces written by Mikhail Kapranov and published by Springer Science & Business Media. This book was released on 2008-03-05 with total page 759 pages. Available in PDF, EPUB and Kindle. Book excerpt: Alexander Reznikov (1960-2003) was a brilliant and highly original mathematician. This book presents 18 articles by prominent mathematicians and is dedicated to his memory. In addition it contains an influential, so far unpublished manuscript by Reznikov of book length. The book further provides an extensive survey on Kleinian groups in higher dimensions and some articles centering on Reznikov as a person.
Download or read book Information Geometry and Its Applications written by Shun-ichi Amari and published by Springer. This book was released on 2016-02-02 with total page 378 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is the first comprehensive book on information geometry, written by the founder of the field. It begins with an elementary introduction to dualistic geometry and proceeds to a wide range of applications, covering information science, engineering, and neuroscience. It consists of four parts, which on the whole can be read independently. A manifold with a divergence function is first introduced, leading directly to dualistic structure, the heart of information geometry. This part (Part I) can be apprehended without any knowledge of differential geometry. An intuitive explanation of modern differential geometry then follows in Part II, although the book is for the most part understandable without modern differential geometry. Information geometry of statistical inference, including time series analysis and semiparametric estimation (the Neyman–Scott problem), is demonstrated concisely in Part III. Applications addressed in Part IV include hot current topics in machine learning, signal processing, optimization, and neural networks. The book is interdisciplinary, connecting mathematics, information sciences, physics, and neurosciences, inviting readers to a new world of information and geometry. This book is highly recommended to graduate students and researchers who seek new mathematical methods and tools useful in their own fields.
Download or read book Mathematical Aspects of String Theory written by Shing-Tung Yau and published by World Scientific Publishing Company Incorporated. This book was released on 1987 with total page 654 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Group Actions in Ergodic Theory Geometry and Topology written by Robert J. Zimmer and published by University of Chicago Press. This book was released on 2019-12-23 with total page 724 pages. Available in PDF, EPUB and Kindle. Book excerpt: Robert J. Zimmer is best known in mathematics for the highly influential conjectures and program that bear his name. Group Actions in Ergodic Theory, Geometry, and Topology: Selected Papers brings together some of the most significant writings by Zimmer, which lay out his program and contextualize his work over the course of his career. Zimmer’s body of work is remarkable in that it involves methods from a variety of mathematical disciplines, such as Lie theory, differential geometry, ergodic theory and dynamical systems, arithmetic groups, and topology, and at the same time offers a unifying perspective. After arriving at the University of Chicago in 1977, Zimmer extended his earlier research on ergodic group actions to prove his cocycle superrigidity theorem which proved to be a pivotal point in articulating and developing his program. Zimmer’s ideas opened the door to many others, and they continue to be actively employed in many domains related to group actions in ergodic theory, geometry, and topology. In addition to the selected papers themselves, this volume opens with a foreword by David Fisher, Alexander Lubotzky, and Gregory Margulis, as well as a substantial introductory essay by Zimmer recounting the course of his career in mathematics. The volume closes with an afterword by Fisher on the most recent developments around the Zimmer program.
Download or read book The Geometry of Fractal Sets written by K. J. Falconer and published by Cambridge University Press. This book was released on 1985 with total page 184 pages. Available in PDF, EPUB and Kindle. Book excerpt: A mathematical study of the geometrical aspects of sets of both integral and fractional Hausdorff dimension. Considers questions of local density, the existence of tangents of such sets as well as the dimensional properties of their projections in various directions.
Download or read book In libri III Decretalium titulos 38 39 40 48 49 50 BSB Clm 27640 written by and published by . This book was released on 1601 with total page 310 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Relativity and Geometry written by Roberto Torretti and published by Elsevier. This book was released on 2014-05-20 with total page 409 pages. Available in PDF, EPUB and Kindle. Book excerpt: Relativity and Geometry aims to elucidate the motivation and significance of the changes in physical geometry brought about by Einstein, in both the first and the second phases of relativity. The book contains seven chapters and a mathematical appendix. The first two chapters review a historical background of relativity. Chapter 3 centers on Einstein's first Relativity paper of 1905. Subsequent chapter presents the Minkowskian formulation of special relativity. Chapters 5 and 6 deal with Einstein's search for general relativity from 1907 to 1915, as well as some aspects and subsequent developments of the theory. The last chapter explores the concept of simultaneity, geometric conventionalism, and a few other questions concerning space time structure, causality, and time.
Download or read book The Principle of Least Action in Geometry and Dynamics written by Karl Friedrich Siburg and published by . This book was released on 2014-01-15 with total page 148 pages. Available in PDF, EPUB and Kindle. Book excerpt:
