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Using numerical integration, it is possible to predict the individual motions of a group of a few celestial objects interacting with each other gravitationally. In this introduction to the few-body problem, a key figure in developing more efficient methods over the past few decades summarizes and explains them, covering both basic analytical formulations and numerical methods. The mathematics required for celestial mechanics and stellar dynamics is explained, starting with two-body motion and progressing through classical methods for planetary system dynamics. This first part of the book can be used as a short course on celestial mechanics. The second part develops the contemporary methods for which the author is renowned - symplectic integration and various methods of regularization. This volume explains the methodology of the subject for graduate students and researchers in celestial mechanics and astronomical dynamics with an interest in few-body dynamics and the regularization of the equations of motion.

Few-body problem. --- Gravitation. --- Celestial mechanics.

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This volume contains the lecture notes of the Third JETSET School on Jets from Young Stars focussing on Numerical MHD and Instabilities. The introductory lectures presented here cover the basic concepts of the numerical methods for the integration of hydrodynamic and magnetohydrodynamic equations and of the applications of these methods to the treatment of the instabilities relevant for the physics of stellar jets. The first part of the book contains an introduction to the finite difference and finite volume methods for computing the solutions of hyperbolic partial differential equations and a discussion of approximate Riemann solvers for both hydrodynamic and magnetohydrodynamic problems. The second part is devoted to the discussion of some of the main instability processes that may take place in stellar jets, namely: the Kelvin-Helmholtz, the radiative shock, the pressure driven and the thermal instabilities. Graduate students and young scientists will benefit from this book by learning how to use the fundamental tools used in computational astrophysical jet research.

Astrophysical jets --- Magnetization instabilities --- Stellar dynamics --- Mathematical models --- Dynamics, Stellar --- Stars --- Celestial mechanics --- Astrophysics --- Jets --- Radio sources (Astronomy) --- Dynamics --- Astrophysics and Astroparticles. --- Numerical and Computational Physics, Simulation. --- Astrophysics. --- Physics. --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Astronomical physics --- Astronomy --- Cosmic physics --- Physics --- Magnetohydrodynamic instabilities --- Hydromagnetic instabilities --- Instabilities, Magnetohydrodynamic --- MHD instabilities --- Magnetohydrodynamics --- Plasma instabilities

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One of the major outstanding questions about black holes is whether they remain stable when subject to small perturbations. An affirmative answer to this question would provide strong theoretical support for the physical reality of black holes. This book takes an important step toward solving the fundamental black hole stability problem in general relativity by establishing the stability of nonrotating black holes - or Schwarzschild spacetimes - under so-called polarized perturbations.

Perturbation (Mathematics) --- Schwarzschild black holes. --- Static black holes --- Black holes (Astronomy) --- Perturbation equations --- Perturbation theory --- Approximation theory --- Dynamics --- Functional analysis --- Mathematical physics --- Perturbation (Astronomy) --- Celestial mechanics --- Bianchi identities. --- Hawking mass. --- Kerr metric. --- Morawetz estimates. --- Reege-Wheeler equations. --- Ricci coefficients. --- Theorem M0. --- asymptotic stability. --- cosmic censorship. --- curvature components. --- decay estimates. --- extreme curvature components. --- general covariance. --- general null frame transformations. --- general theory of relativity. --- geometric analysis. --- invariant quantities. --- mathematical physics, differential geometry. --- molecular orbital theory. --- null structure. --- partial differential equations. --- polarized symmetry. --- space-time.