https://www2.econ.iastate.edu/tesfatsi/DeepLearningInNeuralNetworksOverview.JSchmidhuber2015.pdf

Deep learning in neural networks

We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

A comprehensive review of spatiotemporal pattern formation in systems driven away from equilibrium is presented, with emphasis on comparisons between theory and quantitative experiments. Examples include patterns in hydrodynamic systems such as thermal convection in pure fluids and binary mixtures, Taylor-Couette flow, parametric-wave instabilities, as well as patterns in solidification fronts, nonlinear optics, oscillatory chemical reactions and excitable biological media. The theoretical starting point is usually a set of deterministic equations of motion, typically in the form of nonlinear partial differential equations. These are sometimes supplemented by stochastic terms representing thermal or instrumental noise, but for macroscopic systems and carefully designed experiments the stochastic forces are often negligible. An aim of theory is to describe solutions of the deterministic equations that are likely to be reached starting from typical initial conditions and to persist at long times. A unified description is developed, based on the linear instabilities of a homogeneous state, which leads naturally to a classification of patterns in terms of the characteristic wave vector q0 and frequency ω0 of the instability. Type Is systems (ω0=0, q0≠0) are stationary in time and periodic in space; type IIIo systems (ω0≠0, q0=0) are periodic in time and uniform in space; and type Io systems (ω0≠0, q0≠0) are periodic in both space and time. Near a continuous (or supercritical) instability, the dynamics may be accurately described via "amplitude equations," whose form is universal for each type of instability. The specifics of each system enter only through the nonuniversal coefficients. Far from the instability threshold a different universal description known as the "phase equation" may be derived, but it is restricted to slow distortions of an ideal pattern. For many systems appropriate starting equations are either not known or too complicated to analyze conveniently. It is thus useful to introduce phenomenological order-parameter models, which lead to the correct amplitude equations near threshold, and which may be solved analytically or numerically in the nonlinear regime away from the instability. The above theoretical methods are useful in analyzing "real pattern effects" such as the influence of external boundaries, or the formation and dynamics of defects in ideal structures. An important element in nonequilibrium systems is the appearance of deterministic chaos. A greal deal is known about systems with a small number of degrees of freedom displaying "temporal chaos," where the structure of the phase space can be analyzed in detail. For spatially extended systems with many degrees of freedom, on the other hand, one is dealing with spatiotemporal chaos and appropriate methods of analysis need to be developed. In addition to the general features of nonequilibrium pattern formation discussed above, detailed reviews of theoretical and experimental work on many specific systems are presented. These include Rayleigh-Benard convection in a pure fluid, convection in binary-fluid mixtures, electrohydrodynamic convection in nematic liquid crystals, Taylor-Couette flow between rotating cylinders, parametric surface waves, patterns in certain open flow systems, oscillatory chemical reactions, static and dynamic patterns in biological media, crystallization fronts, and patterns in nonlinear optics. A concluding section summarizes what has and has not been accomplished, and attempts to assess the prospects for the future.

/pdf/pattern-formation-outside-of-equilibrium-1601b9znum.pdf

Pattern formation outside of equilibrium

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications

For billiards in two dimensional domains with boundaries containing only focusing and neutral regular components and satisfacting some geometrical conditionsB-property is proved. Some examples of three and more dimensional domains with billiards obeying this property are also considered.

https://projecteuclid.org/download/pdf_1/euclid.cmp/1103904878

On the ergodic properties of nowhere dispersing billiards

In our previous paper Markov partitions for some classes of dispersed billiards were constructed. Using these partitions we estimate the decay of velocity auto-correlation function and prove the central limit theorem of probability theory and Donsker's Invariance Principle for Lorentz Gas with periodic configuration of scatterers.

/pdf/statistical-properties-of-lorentz-gas-with-periodic-1ua7ezcgeg.pdf

Statistical properties of Lorentz gas with periodic configuration of scatterers

CONTENTS ??1. Introduction ??2. Billiards: necessary information ??3. The Markov lattice: local construction ??4. The Markov lattice: global construction ??5. An estimate for the decay of correlations ??6. The central limit theorem ??7. Applications. Diffusion in deterministic systems Appendix 1 Appendix 2 Appendix 3 References

Statistical properties of two-dimensional hyperbolic billiards

Coupled map lattices have been introduced for studying systems with spatial complexity. The authors consider simple examples of such systems generated by expanding maps of the unit interval (or circle) with some kind of diffusion coupling. It is shown that such systems have a symbolic representation by two-dimensional lattice models of statistical mechanics. The main result states that the Z2 dynamical system generated by space translations and dynamics has a unique invariant mixing Gibbs measure with absolutely continuous finite-dimensional projections. This measure is an analogy of the BRS measure constructed for finite-dimensional hyperbolic transformations.

https://iopscience.iop.org/article/10.1088/0951-7715/1/4/001/pdf

Spacetime chaos in coupled map lattices

Markov Partitions for some classes of billiards in two-dimensional domains on ℝ2 or two-dimensional torus are constructed. Using these partitions we represent the microcanonical distribution of the corresponding dynamical system in the form of a limit Gibbs state and investigate the character of its approximations by finite Markov chains.

/pdf/markov-partitions-for-dispersed-billiards-2w1s28pw0r.pdf

Leonid A. Bunimovich

Papers

On the ergodic properties of nowhere dispersing billiards

Statistical properties of Lorentz gas with periodic configuration of scatterers

Statistical properties of two-dimensional hyperbolic billiards

Spacetime chaos in coupled map lattices

Markov partitions for dispersed billiards