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 the author s gyroalgebra since 1988, the author has 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. Furthermore, Francoise Chatelin noted in her book, and elsewhere, that the computation language that 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 as this book provides new mathematical tools, like hyperbolic simplexes, for the study of hyperbolic geometry in n dimensions. Furthermore, it provides a new look at Einstein s special relativity theory. "

This introductory book uses the moving frame as a tool and develops Finsler geometry on the basis of the Chern connection and the projective sphere bundle. It systematically introduces three classes of geometrical invariants on Finsler manifolds and their intrinsic relations, analyzes local and global results from classic and modern Finsler geometry, and gives non-trivial examples of Finsler manifolds satisfying different curvature conditions.

This book focuses on mesh (grid) enhancement techniques - specifically, the use of selected elliptic methods for both structured and unstructured meshes associated with computational physics applications. Mesh enhancement is the process in which an existing mesh is modified to better meet the requirements of the physics application. To provide the reader with sufficient background information, seven of the nine chapters contain a summary of the numerical simulation process, basic background on mesh terminology and generation approaches, computational geometry, discretization of differential equations, methods of solving linear and nonlinear algebraic systems, geometry of surfaces in Euclidean space, and general elliptic methods for mesh enhancement. Furthermore, these chapters use the concept of harmonic coordinates to develop a unifying framework, the Laplace-Beltrami system, which is the governing principle of the book. The final two chapters apply this scheme, along with other selected elliptic methods, to various structured and unstructured example problems.

This research note presents a complete treatment of the connection between topological circle planes and topological generalized quadrangles. The author uses this connection to provide a better understanding of the relationships between different types of circle planes and to solve a topological version of the problem of Apollonius. Topological Circle Planes and Topological Quadrangles begins with a foundation in classical circle planes and the real symmetric generalized quadrangle and the connection between them. This provides a solid base from which the author offers a more generalized exploration of the topological case. He also compares this treatment to the finite case. Subsequent chapters examine Laguerre, Moebius, and Minkowski planes and their respective relationships to antiregular quadrangles. The author addresses the Lie geometry of each and discuss the relationships of circle planes-the "sisters" of Moebius, Laguerre, and Minkowski planes - and concludes by solving a topological version of the problem of Apollonius in Laguerre, Moebius, and Minkowski planes. The treatment offered in this volume offers complete coverage of the topic. The first part of the text is accessible to anyone with a background in analytic geometry, while the second part requires basic knowledge in general and algebraic topology. Researchers interested in geometry-particularly in topological geometry-will find this volume intriguing and informative. Most of the results presented are new and can be applied to various problems in the field of topological circle planes. Features