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GitHub - lettier/3d-game- shaders-for-beginners: 🎮 A step-by-step guide to implementing SSAO, depth of field, lighting, normal mapping, and more for your 3D game. 🎮 A step-by-step guide to implementing SSAO, depth of field, lighting, normal mapping, and more for your 3D game.GEANT/CLHEP, Release 2.3.1.1, on November 10th, 2015. 5 // MixMax Matrix PseudoRandom Number Generator. Pollicott, Zeta functions and the Periodic Orbit Structure of Hyperbolic Dynamics, Asterisque, Vol. Eckmann, (Springer-Verlag Heidelberg, 1975), 108 pp. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, Vol. Congress of Mathematicians (Stockholm, 1962), pp. Sinai, Probabilistic ideas in ergodic theory, Proc. Knuth, The Art of Computer Programming, Vol. Zhao, Quantum complexity and negative curvature, arXiv:1608.02612.
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The corresponding MIXMAX generator has the best combination of speed, size of the state and is currently available generator. Using analytical and computer calculations, we are studying a distribution function of periodic trajectories and their deviation from asymptotic behavior. The asymptotic distribution of chaotic trajectories of C-K systems with periods less than a given number is well known in mathematical literature, but a deviation from its asymptotic behavior is unknown. The C-K systems on a torus have countable set of everywhere dense periodic trajectories and their distribution play a crucial role in coding and implementation of the pseudorandom number generator.
All trajectories of the C-K systems are exponentially unstable and pseudorandom numbers are represented in terms of coordinates of very long chaotic trajectories. We are considering the hyperbolic C-K systems of Anosov–Kolmogorov which are defined on high dimensional tori and are used to generate pseudorandom numbers for Monte-Carlo simulations.
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