Abraham Albert Ungar – författare

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6 produkter

Analytic Hyperbolic Geometry in N Dimensions
An Introduction
AvAbraham Albert Ungar
Inbunden, Engelska, 2014
2 995 kr
Skickas inom 10-15 vardagar
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.
Analytic Hyperbolic Geometry in N Dimensions
An Introduction
AvAbraham Albert Ungar
E-bok
PDF, Engelska, 2014
3 558 kr
Läs direkt efter köp
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 gyroalgebraÂ in 1988, the author crafted gyrolanguage, the algebraic language t
ANALYTIC HYPERBOLIC GEOMETRY AND ALBERT EINSTEIN'S SPECIAL THEORY OF RELATIVITY (SECOND EDITION)
AvAbraham Albert Ungar
Inbunden, Engelska, 2022
2 703 kr
Skickas inom 3-6 vardagar
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.
BARYCENTRIC CALCULUS IN EUCLIDEAN AND HYPERBOLIC GEOMETRY: A COMPARATIVE INTRODUCTION
AvAbraham Albert Ungar
Inbunden, Engelska, 2010
1 737 kr
Skickas inom 5-8 vardagar
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.
BARYCENTRIC CALCULUS IN EUCLIDEAN AND HYPERBOLIC GEOMETRY: A COMPARATIVE INTRODUCTION (SECOND EDITION)
AvAbraham Albert Ungar
Inbunden, Engelska, 2025
1 909 kr
Skickas inom 3-6 vardagar
This unique and richly illustrated book explores barycentric calculus, a geometric method grounded in the concept of the center of gravity. Used to elegantly determine triangle centers through weighted points, barycentric coordinates have long revealed deep insights in Euclidean geometry. Now, this book extends those insights to the fascinating realm of hyperbolic geometry, building a powerful bridge between classical and modern mathematical worlds.In Euclidean geometry, over 3,000 triangle centers have been identified using barycentric coordinates. This book introduces readers to their hyperbolic analogs, uncovering remarkable parallels between triangle centers in Bolyai–Lobachevsky geometry and their Euclidean counterparts. The author's innovative use of Cartesian coordinates, trigonometry, and vector algebra — adapted for hyperbolic geometry — equips readers with familiar yet powerful tools to explore unfamiliar terrain.At the heart of the book is the development of hyperbolic barycentric coordinates, or gyrobarycentric coordinates, within the framework of gyrovector spaces — a novel algebraic structure emerging from Einstein's velocity addition and Möbius addition. These gyrovectors underpin the Klein and Poincaré ball models of hyperbolic geometry, just as traditional vectors underlie analytic Euclidean geometry.Whether you are a researcher in geometry, mathematical physics, or relativity, or simply fascinated by the deep structure of space, this book offers a groundbreaking approach to analytic hyperbolic geometry through barycentric and gyrobarycentric coordinates.
ANALYTIC HYPERBOLIC GEOMETRY AND ALBERT EINSTEIN'S SPECIAL THEORY OF RELATIVITY
AvAbraham Albert Ungar
Inbunden, Engelska, 2008
3 875 kr
Tillfälligt slut
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. It introduces the notion of vectors into analytic hyperbolic geometry, where they are called gyrovectors.Newtonian velocity addition is the common vector addition, which is both commutative and associative. The resulting vector spaces, in turn, form the algebraic setting for the standard model of Euclidean geometry. In full analogy, Einsteinian velocity addition is a gyrovector addition, which is both gyrocommutative and gyroassociative. The resulting gyrovector spaces, in turn, form the algebraic setting for the Beltrami-Klein ball model of the hyperbolic geometry of Bolyai and Lobachevsky. Similarly, Möbius addition gives rise to gyrovector spaces that form the algebraic setting for the Poincaré ball model of hyperbolic geometry.In full analogy with classical results, the book presents a novel relativistic interpretation of stellar aberration in terms of relativistic gyrotrigonometry and gyrovector addition. 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. The novel relativistic resultant mass of the system, concentrated at the relativistic center of mass, dictates the validity of the dark matter and the dark energy that were introduced by cosmologists as ad hoc postulates to explain cosmological observations about missing gravitational force and late-time cosmic accelerated expansion.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 analytic hyperbolic geometry.

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Abraham Albert Ungar – författare

Analytic Hyperbolic Geometry in N Dimensions

Analytic Hyperbolic Geometry in N Dimensions

ANALYTIC HYPERBOLIC GEOMETRY AND ALBERT EINSTEIN'S SPECIAL THEORY OF RELATIVITY (SECOND EDITION)

BARYCENTRIC CALCULUS IN EUCLIDEAN AND HYPERBOLIC GEOMETRY: A COMPARATIVE INTRODUCTION

BARYCENTRIC CALCULUS IN EUCLIDEAN AND HYPERBOLIC GEOMETRY: A COMPARATIVE INTRODUCTION (SECOND EDITION)

ANALYTIC HYPERBOLIC GEOMETRY AND ALBERT EINSTEIN'S SPECIAL THEORY OF RELATIVITY

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