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Geodesics (Mathematics). --- Geodesy

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Riemannian manifolds --- Curves on surfaces --- Geodesics (Mathematics) --- Riemann, Variétés de --- Courbes sur les surfaces --- Curves on surfaces. --- Riemannian manifolds. --- Geodesics (Mathematics). --- Riemann, Variétés de --- 514.7 --- 514.7 Differential geometry. Algebraic and analytic methods in geometry --- Differential geometry. Algebraic and analytic methods in geometry --- Differential geometry. Global analysis --- Geodesy. Cartography --- Géométrie différentielle globale. --- Géodésiques (mathématiques) --- Global differential geometry --- Géométrie différentielle globale. --- Géodésiques (mathématiques)

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Despite its importance in the history of Ancient science, Menelaus’ Spherics is still by and large unknown. This treatise, which lies at the foundation of spherical geometry, is lost in Greek but has been preserved in its Arabic versions. The reader will find here, for the first time edited and translated into English, the essentials of this tradition, namely: a fragment of an early Arabic translation and the first Arabic redaction of the Spherics composed by al-Māhānī /al-Harawī, together with a historical and mathematical study of Menelaus’ treatise. With this book, a new and important part of the Greek and Arabic legacy to the history of mathematics comes to light. This book will be an indispensable acquisition for any reader interested in the history of Ancient geometry and science and, more generally, in Greek and Arabic science and culture.

Geodesics (Mathematics) --- Sphere. --- Geometry, Differential. --- Curves on surfaces --- Surfaces --- Curved surfaces --- Geometry --- Shapes --- Surfaces, Curves on --- Differential geometry --- Geometry, Solid --- Orbs --- Geometry, Differential --- Global analysis (Mathematics) --- Mathematics --- Mathematical models. --- Menelaus, --- مانالاوس --- Mānālāwus --- Ménélaos, --- Menélaus, --- Millaeus, --- Ménélas, --- Ménélaüs, --- Menelao, --- Menelaos, --- Arabic mathematics. --- Greek geometry. --- History of non-Euclidean geometry. --- Spherics.

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This book provides a complete exposition of equidistribution and counting problems weighted by a potential function of common perpendicular geodesics in negatively curved manifolds and simplicial trees. Avoiding any compactness assumptions, the authors extend the theory of Patterson-Sullivan, Bowen-Margulis and Oh-Shah (skinning) measures to CAT(-1) spaces with potentials. The work presents a proof for the equidistribution of equidistant hypersurfaces to Gibbs measures, and the equidistribution of common perpendicular arcs between, for instance, closed geodesics. Using tools from ergodic theory (including coding by topological Markov shifts, and an appendix by Buzzi that relates weak Gibbs measures and equilibrium states for them), the authors further prove the variational principle and rate of mixing for the geodesic flow on metric and simplicial trees—again without the need for any compactness or torsionfree assumptions. In a series of applications, using the Bruhat-Tits trees over non-Archimedean local fields, the authors subsequently prove further important results: the Mertens formula and the equidistribution of Farey fractions in function fields, the equidistribution of quadratic irrationals over function fields in their completions, and asymptotic counting results of the representations by quadratic norm forms. One of the book's main benefits is that the authors provide explicit error terms throughout. Given its scope, it will be of interest to graduate students and researchers in a wide range of fields, for instance ergodic theory, dynamical systems, geometric group theory, discrete subgroups of locally compact groups, and the arithmetic of function fields.

Geodesics (Mathematics) --- Geometry, Differential --- Global analysis (Mathematics) --- Mathematics --- Dynamics. --- Ergodic theory. --- Differential geometry. --- Group theory. --- Number theory. --- Convex geometry . --- Discrete geometry. --- Probabilities. --- Dynamical Systems and Ergodic Theory. --- Differential Geometry. --- Group Theory and Generalizations. --- Number Theory. --- Convex and Discrete Geometry. --- Probability Theory and Stochastic Processes. --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Discrete mathematics --- Geometry --- Combinatorial geometry --- Number study --- Numbers, Theory of --- Algebra --- Groups, Theory of --- Substitutions (Mathematics) --- Differential geometry --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics