Download or read book Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession written by Abraham A. Ungar and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 449 pages. Available in PDF, EPUB and Kindle. Book excerpt: "I cannot define coincidence [in mathematics]. But 1 shall argue that coincidence can always be elevated or organized into a superstructure which perfonns a unification along the coincidental elements. The existence of a coincidence is strong evidence for the existence of a covering theory. " -Philip 1. Davis [Dav81] Alluding to the Thomas gyration, this book presents the Theory of gy rogroups and gyrovector spaces, taking the reader to the immensity of hyper bolic geometry that lies beyond the Einstein special theory of relativity. Soon after its introduction by Einstein in 1905 [Ein05], special relativity theory (as named by Einstein ten years later) became overshadowed by the ap pearance of general relativity. Subsequently, the exposition of special relativity followed the lines laid down by Minkowski, in which the role of hyperbolic ge ometry is not emphasized. This can doubtlessly be explained by the strangeness and unfamiliarity of hyperbolic geometry [Bar98]. The aim of this book is to reverse the trend of neglecting the role of hy perbolic geometry in the special theory of relativity, initiated by Minkowski, by emphasizing the central role that hyperbolic geometry plays in the theory.

Download or read book Beyond the Einstein Addition Law and Its Gyroscopic Thomas Precession written by Abraham A Ungar and published by . This book was released on 2001-03-12 with total page 464 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession written by A.A. Ungar and published by Springer. This book was released on 2001-11-30 with total page 464 pages. Available in PDF, EPUB and Kindle. Book excerpt: Evidence that Einstein's addition is regulated by the Thomas precession has come to light, turning the notorious Thomas precession, previously considered the ugly duckling of special relativity theory, into the beautiful swan of gyrogroup and gyrovector space theory, where it has been extended by abstraction into an automorphism generator, called the Thomas gyration. The Thomas gyration, in turn, allows the introduction of vectors into hyperbolic geometry, where they are called gyrovectors, in such a way that Einstein's velocity additions turns out to be a gyrovector addition. Einstein's addition thus becomes a gyrocommutative, gyroassociative gyrogroup operation in the same way that ordinary vector addition is a commutative, associative group operation. Some gyrogroups of gyrovectors admit scalar multiplication, giving rise to gyrovector spaces in the same way that some groups of vectors that admit scalar multiplication give rise to vector spaces. Furthermore, gyrovector spaces form the setting for hyperbolic geometry in the same way that vector spaces form the setting for Euclidean geometry. In particular, the gyrovector space with gyrovector addition given by Einstein's (Möbius') addition forms the setting for the Beltrami (Poincaré) ball model of hyperbolic geometry. The gyrogroup-theoretic techniques developed in this book for use in relativity physics and in hyperbolic geometry allow one to solve old and new important problems in relativity physics. A case in point is Einstein's 1905 view of the Lorentz length contraction, which was contradicted in 1959 by Penrose, Terrell and others. The application of gyrogroup-theoretic techniques clearly tilt the balance in favor of Einstein.

Download or read book Beyond Pseudo Rotations in Pseudo Euclidean Spaces written by Abraham Ungar and published by Academic Press. This book was released on 2018-01-10 with total page 420 pages. Available in PDF, EPUB and Kindle. Book excerpt: Beyond Pseudo-Rotations in Pseudo-Euclidean Spaces presents for the first time a unified study of the Lorentz transformation group SO(m, n) of signature (m, n), m, n ? N, which is fully analogous to the Lorentz group SO(1, 3) of Einstein's special theory of relativity. It is based on a novel parametric realization of pseudo-rotations by a vector-like parameter with two orientation parameters. The book is of interest to specialized researchers in the areas of algebra, geometry and mathematical physics, containing new results that suggest further exploration in these areas. - Introduces the study of generalized gyrogroups and gyrovector spaces - Develops new algebraic structures, bi-gyrogroups and bi-gyrovector spaces - Helps readers to surmount boundaries between algebra, geometry and physics - Assists readers to parametrize and describe the full set of generalized Lorentz transformations in a geometric way - Generalizes approaches from gyrogroups and gyrovector spaces to bi-gyrogroups and bi-gyrovector spaces with geometric entanglement

Download or read book Hyperbolic Triangle Centers written by A.A. Ungar and published by Springer Science & Business Media. This book was released on 2010-06-18 with total page 322 pages. Available in PDF, EPUB and Kindle. Book excerpt: After A. Ungar had introduced vector algebra and Cartesian coordinates into hyperbolic geometry in his earlier books, along with novel applications in Einstein’s special theory of relativity, the purpose of his new book is to introduce hyperbolic barycentric coordinates, another important concept to embed Euclidean geometry into hyperbolic geometry. It will be demonstrated that, in full analogy to classical mechanics where barycentric coordinates are related to the Newtonian mass, barycentric coordinates are related to the Einsteinian relativistic mass in hyperbolic geometry. Contrary to general belief, Einstein’s relativistic mass hence meshes up extraordinarily well with Minkowski’s four-vector formalism of special relativity. In Euclidean geometry, barycentric coordinates can be used to determine various triangle centers. While there are many known Euclidean triangle centers, only few hyperbolic triangle centers are known, and none of the known hyperbolic triangle centers has been determined analytically with respect to its hyperbolic triangle vertices. In his recent research, the author set the ground for investigating hyperbolic triangle centers via hyperbolic barycentric coordinates, and one of the purposes of this book is to initiate a study of hyperbolic triangle centers in full analogy with the rich study of Euclidean triangle centers. Owing to its novelty, the book is aimed at a large audience: it can be enjoyed equally by upper-level undergraduates, graduate students, researchers and academics in geometry, abstract algebra, theoretical physics and astronomy. For a fruitful reading of this book, familiarity with Euclidean geometry is assumed. Mathematical-physicists and theoretical physicists are likely to enjoy the study of Einstein’s special relativity in terms of its underlying hyperbolic geometry. Geometers may enjoy the hunt for new hyperbolic triangle centers and, finally, astronomers may use hyperbolic barycentric coordinates in the velocity space of cosmology.

Download or read book Analytic Hyperbolic Geometry And Albert Einstein s Special Theory Of Relativity Second Edition written by Abraham Albert Ungar and published by World Scientific. This book was released on 2022-02-22 with total page 775 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents a powerful way to study Einstein's special theory of relativity and its underlying hyperbolic geometry in which analogies with classical results form the right tool. The premise of analogy as a study strategy is to make the unfamiliar familiar. Accordingly, this book introduces the notion of vectors into analytic hyperbolic geometry, where they are called gyrovectors. Gyrovectors turn out to be equivalence classes that add according to the gyroparallelogram law just as vectors are equivalence classes that add according to the parallelogram law. In the gyrolanguage of this book, accordingly, one prefixes a gyro to a classical term to mean the analogous term in hyperbolic geometry. As an example, the relativistic gyrotrigonometry of Einstein's special relativity is developed and employed to the study of the stellar aberration phenomenon in astronomy.Furthermore, the book presents, for the first time, the relativistic center of mass of an isolated system of noninteracting particles that coincided at some initial time t = 0. It turns out that the invariant mass of the relativistic center of mass of an expanding system (like galaxies) exceeds the sum of the masses of its constituent particles. This excess of mass suggests a viable mechanism for the formation of dark matter in the universe, which has not been detected but is needed to gravitationally 'glue' each galaxy in the universe. The discovery of the relativistic center of mass in this book thus demonstrates once again the usefulness of the study of Einstein's special theory of relativity in terms of its underlying hyperbolic geometry.

Download or read book Essays in Mathematics and its Applications written by Panos M. Pardalos and published by Springer Science & Business Media. This book was released on 2012-08-07 with total page 501 pages. Available in PDF, EPUB and Kindle. Book excerpt: ​The volume is dedicated to Stephen Smale on the occasion of his 80th birthday.Besides his startling 1960 result of the proof of the Poincar ́e conjecture for all dimensionsgreater than or equal to five, Smale’s ground breaking contributions invarious fields in Mathematics have marked the second part of the 20th century andbeyond. Stephen Smale has done pioneering work in differential topology, globalanalysis, dynamical systems, nonlinear functional analysis, numerical analysis, theoryof computation and machine learning as well as applications in the physical andbiological sciences and economics. In sum, Stephen Smale has manifestly brokenthe barriers among the different fields of mathematics and dispelled some remainingprejudices. He is indeed a universal mathematician. Smale has been honoredwith several prizes and honorary degrees including, among others, the Fields Medal(1966), The Veblen Prize (1966), the National Medal of Science (1996) and theWolfPrize (2006/2007).

Download or read book Nonlinear Analysis written by Panos M. Pardalos and published by Springer Science & Business Media. This book was released on 2012-06-02 with total page 898 pages. Available in PDF, EPUB and Kindle. Book excerpt: The volume will consist of about 40 articles written by some very influential mathematicians of our time and will expose the latest achievements in the broad area of nonlinear analysis and its various interdisciplinary applications.

Download or read book Non Euclidean Geometries written by András Prékopa and published by Springer Science & Business Media. This book was released on 2006-06-03 with total page 497 pages. Available in PDF, EPUB and Kindle. Book excerpt: "From nothing I have created a new different world," wrote János Bolyai to his father, Wolgang Bolyai, on November 3, 1823, to let him know his discovery of non-Euclidean geometry, as we call it today. The results of Bolyai and the co-discoverer, the Russian Lobachevskii, changed the course of mathematics, opened the way for modern physical theories of the twentieth century, and had an impact on the history of human culture. The papers in this volume, which commemorates the 200th anniversary of the birth of János Bolyai, were written by leading scientists of non-Euclidean geometry, its history, and its applications. Some of the papers present new discoveries about the life and works of János Bolyai and the history of non-Euclidean geometry, others deal with geometrical axiomatics; polyhedra; fractals; hyperbolic, Riemannian and discrete geometry; tilings; visualization; and applications in physics.

Download or read book Essays in Mathematics and its Applications written by Themistocles M. Rassias and published by Springer. This book was released on 2016-06-14 with total page 659 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume, dedicated to the eminent mathematician Vladimir Arnold, presents a collection of research and survey papers written on a large spectrum of theories and problems that have been studied or introduced by Arnold himself. Emphasis is given to topics relating to dynamical systems, stability of integrable systems, algebraic and differential topology, global analysis, singularity theory and classical mechanics. A number of applications of Arnold’s groundbreaking work are presented. This publication will assist graduate students and research mathematicians in acquiring an in-depth understanding and insight into a wide domain of research of an interdisciplinary nature.

Download or read book Analytic Hyperbolic Geometry in N Dimensions written by Abraham Albert Ungar and published by CRC Press. This book was released on 2014-12-17 with total page 623 pages. Available in PDF, EPUB and Kindle. Book excerpt: The concept of the Euclidean simplex is important in the study of n-dimensional Euclidean geometry. This book introduces for the first time the concept of hyperbolic simplex as an important concept in n-dimensional hyperbolic geometry. Following the emergence of his gyroalgebra in 1988, the author crafted gyrolanguage, the algebraic language that sheds natural light on hyperbolic geometry and special relativity. Several authors have successfully employed the author’s gyroalgebra in their exploration for novel results. Françoise Chatelin noted in her book, and elsewhere, that the computation language of Einstein described in this book plays a universal computational role, which extends far beyond the domain of special relativity. This book will encourage researchers to use the author’s novel techniques to formulate their own results. The book provides new mathematical tools, such as hyperbolic simplexes, for the study of hyperbolic geometry in n dimensions. It also presents a new look at Einstein’s special relativity theory.

Download or read book A Gyrovector Space Approach to Hyperbolic Geometry written by Abraham Ungar and published by Springer Nature. This book was released on 2022-06-01 with total page 182 pages. Available in PDF, EPUB and Kindle. Book excerpt: The mere mention of hyperbolic geometry is enough to strike fear in the heart of the undergraduate mathematics and physics student. Some regard themselves as excluded from the profound insights of hyperbolic geometry so that this enormous portion of human achievement is a closed door to them. The mission of this book is to open that door by making the hyperbolic geometry of Bolyai and Lobachevsky, as well as the special relativity theory of Einstein that it regulates, accessible to a wider audience in terms of novel analogies that the modern and unknown share with the classical and familiar. These novel analogies that this book captures stem from Thomas gyration, which is the mathematical abstraction of the relativistic effect known as Thomas precession. Remarkably, the mere introduction of Thomas gyration turns Euclidean geometry into hyperbolic geometry, and reveals mystique analogies that the two geometries share. Accordingly, Thomas gyration gives rise to the prefix "gyro" that is extensively used in the gyrolanguage of this book, giving rise to terms like gyrocommutative and gyroassociative binary operations in gyrogroups, and gyrovectors in gyrovector spaces. Of particular importance is the introduction of gyrovectors into hyperbolic geometry, where they are equivalence classes that add according to the gyroparallelogram law in full analogy with vectors, which are equivalence classes that add according to the parallelogram law. A gyroparallelogram, in turn, is a gyroquadrilateral the two gyrodiagonals of which intersect at their gyromidpoints in full analogy with a parallelogram, which is a quadrilateral the two diagonals of which intersect at their midpoints. Table of Contents: Gyrogroups / Gyrocommutative Gyrogroups / Gyrovector Spaces / Gyrotrigonometry

Download or read book Analytic Hyperbolic Geometry written by Abraham A. Ungar and published by World Scientific. This book was released on 2005 with total page 484 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is the first book on analytic hyperbolic geometry, fully analogous to analytic Euclidean geometry. Analytic hyperbolic geometry regulates relativistic mechanics just as analytic Euclidean geometry regulates classical mechanics. The book presents a novel gyrovector space approach to analytic hyperbolic geometry, fully analogous to the well-known vector space approach to Euclidean geometry. A gyrovector is a hyperbolic vector. In the resulting "gyrolanguage" of the book, one attaches the prefix "gyro" to a classical term to mean the analogous term in hyperbolic geometry. The book begins with the definition of gyrogroups, which is fully analogous to the definition of groups. Gyrogroups, both gyrocommutative and nongyrocommutative, abound in group theory. Surprisingly, the seemingly structureless Einstein velocity addition of special relativity turns out to be a gyrocommutative gyrogroup operation. Introducing scalar multiplication, some gyrocommutative gyrogroups of gyrovectors become gyrovector spaces. The latter, in turn, form the setting for analytic hyperbolic geometry just as vector spaces form the setting for analytic Euclidean geometry. By hybrid techniques of differential geometry and gyrovector spaces, it is shown that Einstein (Mobius) gyrovector spaces form the setting for Beltrami-Klein (Poincare) ball models of hyperbolic geometry. Finally, novel applications of Mobius gyrovector spaces in quantum computation, and of Einstein gyrovector spaces in special relativity, are presented.

Download or read book Hypercomplex Analysis and Applications written by Irene Sabadini and published by Springer Science & Business Media. This book was released on 2010-12-20 with total page 280 pages. Available in PDF, EPUB and Kindle. Book excerpt: The purpose of the volume is to bring forward recent trends of research in hypercomplex analysis. The list of contributors includes first rate mathematicians and young researchers working on several different aspects in quaternionic and Clifford analysis. Besides original research papers, there are papers providing the state-of-the-art of a specific topic, sometimes containing interdisciplinary fields. The intended audience includes researchers, PhD students, postgraduate students who are interested in the field and in possible connection between hypercomplex analysis and other disciplines, including mathematical analysis, mathematical physics, algebra.

Download or read book Barycentric Calculus in Euclidean and Hyperbolic Geometry written by Abraham A. Ungar and published by World Scientific. This book was released on 2010 with total page 360 pages. Available in PDF, EPUB and Kindle. Book excerpt: The word barycentric is derived from the Greek word barys (heavy), and refers to center of gravity. Barycentric calculus is a method of treating geometry by considering a point as the center of gravity of certain other points to which weights are ascribed. Hence, in particular, barycentric calculus provides excellent insight into triangle centers. This unique book on barycentric calculus in Euclidean and hyperbolic geometry provides an introduction to the fascinating and beautiful subject of novel triangle centers in hyperbolic geometry along with analogies they share with familiar triangle centers in Euclidean geometry. As such, the book uncovers magnificent unifying notions that Euclidean and hyperbolic triangle centers share. In his earlier books the author adopted Cartesian coordinates, trigonometry and vector algebra for use in hyperbolic geometry that is fully analogous to the common use of Cartesian coordinates, trigonometry and vector algebra in Euclidean geometry. As a result, powerful tools that are commonly available in Euclidean geometry became available in hyperbolic geometry as well, enabling one to explore hyperbolic geometry in novel ways. In particular, this new book establishes hyperbolic barycentric coordinates that are used to determine various hyperbolic triangle centers just as Euclidean barycentric coordinates are commonly used to determine various Euclidean triangle centers. The hunt for Euclidean triangle centers is an old tradition in Euclidean geometry, resulting in a repertoire of more than three thousand triangle centers that are known by their barycentric coordinate representations. The aim of this book is to initiate a fully analogous hunt for hyperbolic triangle centers that will broaden the repertoire of hyperbolic triangle centers provided here.

Download or read book Mathematics Without Boundaries written by Themistocles M. Rassias and published by Springer. This book was released on 2014-09-17 with total page 783 pages. Available in PDF, EPUB and Kindle. Book excerpt: The contributions in this volume have been written by eminent scientists from the international mathematical community and present significant advances in several theories, methods and problems of Mathematical Analysis, Discrete Mathematics, Geometry and their Applications. The chapters focus on both old and recent developments in Functional Analysis, Harmonic Analysis, Complex Analysis, Operator Theory, Combinatorics, Functional Equations, Differential Equations as well as a variety of Applications. The book also contains some review works, which could prove particularly useful for a broader audience of readers in Mathematical Sciences, and especially to graduate students looking for the latest information.

Download or read book Advanced Mathematical Techniques in Computational and Intelligent Systems written by Sandeep Singh and published by CRC Press. This book was released on 2023-11-20 with total page 285 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book comprehensively discusses the modeling of real-world industrial problems and innovative optimization techniques such as heuristics, finite methods, operation research techniques, intelligent algorithms, and agent- based methods. Discusses advanced techniques such as key cell, Mobius inversion, and zero suffix techniques to find initial feasible solutions to optimization problems. Provides a useful guide toward the development of a sustainable model for disaster management. Presents optimized hybrid block method techniques to solve mathematical problems existing in the industries. Covers mathematical techniques such as Laplace transformation, stochastic process, and differential techniques related to reliability theory. Highlights application on smart agriculture, smart healthcare, techniques for disaster management, and smart manufacturing. Advanced Mathematical Techniques in Computational and Intelligent Systems is primarily written for graduate and senior undergraduate students, as well as academic researchers in electrical engineering, electronics and communications engineering, computer engineering, and mathematics.