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This monograph presents recent developments of the theory of algebraic dynamical systems and their applications to computer sciences, cryptography, cognitive sciences, psychology, image analysis, and numerical simulations. The most important mathematical results presented in this book are in the fields of ergodicity, p-adic numbers, and noncommutative groups. For students and researchers working on the theory of dynamical systems, algebra, number theory, measure theory, computer sciences, cryptography, and image analysis.

Arithmetical algebraic geometry. --- Differentiable dynamical systems. --- Algebraic Dynamical Systems. --- Automata. --- Computer Sciences. --- Cryptography. --- Ergodicity. --- Genetics. --- Latin Squares. --- Measure-Preserving Transformations. --- Profinite Groups. --- Pseudorandom Generators. --- Psychology. --- Quantum Theory. --- Stream Ciphers. --- p-adic Numbers. --- p-adic Probability.

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Metastable Liquids provides a comprehensive treatment of the properties of liquids under conditions where the stable state is a vapor, a solid, or a liquid mixture of different composition. It examines the fundamental principles that govern the equilibrium properties, stability, relaxation mechanisms, and relaxation rates of metastable liquids. Building on the interplay of kinetics and thermodynamics that determines the thermophysical properties and structural relaxation of metastable liquids, it offers an in-depth treatment of thermodynamic stability theory, the statistical mechanics of metastability, nucleation, spinodal decomposition, supercooled liquids, and the glass transition. Both traditional topics--such as stability theory--and modern developments--including modern theories of nucleation and the properties of supercooled and glassy water--are treated in detail. An introductory chapter illustrates, with numerous examples, the importance and ubiquity of metastable liquids. Examples include the ascent of sap in plants, the strategies adopted by many living organisms to survive prolonged exposure to sub-freezing conditions, the behavior of proteins at low temperatures, metastability in mineral inclusions, ozone depletion, the preservation and storage of labile biochemicals, and the prevention of natural gas clathrate hydrate formation. All mathematical symbols are defined in the text and key equations are clearly explained. More complex mathematical explanations are available in the appendixes.

Liquids --- Supercooled liquids. --- Phase transformations (Statistical physics) --- Chemistry, Physical and theoretical. --- Thermal properties. --- Chemistry, Theoretical --- Physical chemistry --- Theoretical chemistry --- Chemistry --- Phase changes (Statistical physics) --- Phase transitions (Statistical physics) --- Phase rule and equilibrium --- Statistical physics --- Supercooled fluids --- Adam-Gibbs theory. --- Ginzburg criterion. --- Kauzmann paradox. --- Laplace equation. --- Maxwell construction. --- antifreeze proteins. --- beta relaxation. --- capillarity approximation. --- critical nucleus. --- crystallization. --- detailed balance. --- dividing surface. --- embryos. --- energetics of formation. --- fragile liquids. --- free energy barrier. --- hard sphere fluid. --- ice. --- interfacial tension. --- mean-field theory. --- non-ergodicity parameter. --- order parameter. --- polyamorphism. --- spinodal curve. --- structural arrest. --- supercooled vapors. --- temperature.

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Locally symmetric spaces are generalizations of spaces of constant curvature. In this book the author presents the proof of a remarkable phenomenon, which he calls "strong rigidity": this is a stronger form of the deformation rigidity that has been investigated by Selberg, Calabi-Vesentini, Weil, Borel, and Raghunathan.The proof combines the theory of semi-simple Lie groups, discrete subgroups, the geometry of E. Cartan's symmetric Riemannian spaces, elements of ergodic theory, and the fundamental theorem of projective geometry as applied to Tit's geometries. In his proof the author introduces two new notions having independent interest: one is "pseudo-isometries"; the other is a notion of a quasi-conformal mapping over the division algebra K (K equals real, complex, quaternion, or Cayley numbers). The author attempts to make the account accessible to readers with diverse backgrounds, and the book contains capsule descriptions of the various theories that enter the proof.

Differential geometry. Global analysis --- Riemannian manifolds --- Symmetric spaces --- Rigidity (Geometry) --- 512 --- Lie groups --- Geometric rigidity --- Rigidity theorem --- Discrete geometry --- Spaces, Symmetric --- Geometry, Differential --- Manifolds, Riemannian --- Riemannian space --- Space, Riemannian --- Manifolds (Mathematics) --- Groups, Lie --- Lie algebras --- Topological groups --- Algebra --- Lie groups. --- Riemannian manifolds. --- Symmetric spaces. --- Rigidity (Geometry). --- 512 Algebra --- Addition. --- Adjoint representation. --- Affine space. --- Approximation. --- Automorphism. --- Axiom. --- Big O notation. --- Boundary value problem. --- Cohomology. --- Compact Riemann surface. --- Compact space. --- Conjecture. --- Constant curvature. --- Corollary. --- Counterexample. --- Covering group. --- Covering space. --- Curvature. --- Diameter. --- Diffeomorphism. --- Differentiable function. --- Dimension. --- Direct product. --- Division algebra. --- Ergodicity. --- Erlangen program. --- Existence theorem. --- Exponential function. --- Finitely generated group. --- Fundamental domain. --- Fundamental group. --- Geometry. --- Half-space (geometry). --- Hausdorff distance. --- Hermitian matrix. --- Homeomorphism. --- Homomorphism. --- Hyperplane. --- Identity matrix. --- Inner automorphism. --- Isometry group. --- Jordan algebra. --- Matrix multiplication. --- Metric space. --- Morphism. --- Möbius transformation. --- Normal subgroup. --- Normalizing constant. --- Partially ordered set. --- Permutation. --- Projective space. --- Riemann surface. --- Riemannian geometry. --- Sectional curvature. --- Self-adjoint. --- Set function. --- Smoothness. --- Stereographic projection. --- Subgroup. --- Subset. --- Summation. --- Symmetric space. --- Tangent space. --- Tangent vector. --- Theorem. --- Topology. --- Tubular neighborhood. --- Two-dimensional space. --- Unit sphere. --- Vector group. --- Weyl group. --- Riemann, Variétés de --- Lie, Groupes de --- Geometrie differentielle globale --- Varietes riemanniennes

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It is very well known that differential equations are related with the rise of physical science in the last several decades and they are used successfully for models of real-world problems in a variety of fields from several disciplines. Additionally, difference equations represent the discrete analogues of differential equations. These types of equations started to be used intensively during the last several years for their multiple applications, particularly in complex chaotic behavior. A certain class of differential and related difference equations is represented by their respective fractional forms, which have been utilized to better describe non-local phenomena appearing in all branches of science and engineering. The purpose of this book is to present some common results given by mathematicians together with physicists, engineers, as well as other scientists, for whom differential and difference equations are valuable research tools. The reported results can be used by researchers and academics working in both pure and applied differential equations.

Research & information: general --- Mathematics & science --- dynamic equations --- time scales --- classification --- existence --- necessary and sufficient conditions --- fractional calculus --- triangular fuzzy number --- double-parametric form --- FRDTM --- fractional dynamical model of marriage --- approximate controllability --- degenerate evolution equation --- fractional Caputo derivative --- sectorial operator --- fractional symmetric Hahn integral --- fractional symmetric Hahn difference operator --- Arrhenius activation energy --- rotating disk --- Darcy–Forchheimer flow --- binary chemical reaction --- nanoparticles --- numerical solution --- fractional differential equations --- two-dimensional wavelets --- finite differences --- fractional diffusion-wave equation --- fractional derivative --- ill-posed problem --- Tikhonov regularization method --- non-linear differential equation --- cubic B-spline --- central finite difference approximations --- absolute errors --- second order differential equations --- mild solution --- non-instantaneous impulses --- Kuratowski measure of noncompactness --- Darbo fixed point --- multi-stage method --- multi-step method --- Runge–Kutta method --- backward difference formula --- stiff system --- numerical solutions --- Riemann-Liouville fractional integral --- Caputo fractional derivative --- fractional Taylor vector --- kerosene oil-based fluid --- stagnation point --- carbon nanotubes --- variable thicker surface --- thermal radiation --- differential equations --- symmetric identities --- degenerate Hermite polynomials --- complex zeros --- oscillation --- third order --- mixed neutral differential equations --- powers of stochastic Gompertz diffusion models --- powers of stochastic lognormal diffusion models --- estimation in diffusion process --- stationary distribution and ergodicity --- trend function --- application to simulated data --- n-th order linear differential equation --- two-point boundary value problem --- Green function --- linear differential equation --- exponential stability --- linear output feedback --- stabilization --- uncertain system --- nonlocal effects --- linear control system --- Hilbert space --- state feedback control --- exact controllability --- upper Bohl exponent

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Extremely popular for statistical inference, Bayesian methods are also becoming popular in machine learning and artificial intelligence problems. Bayesian estimators are often implemented by Monte Carlo methods, such as the Metropolis–Hastings algorithm of the Gibbs sampler. These algorithms target the exact posterior distribution. However, many of the modern models in statistics are simply too complex to use such methodologies. In machine learning, the volume of the data used in practice makes Monte Carlo methods too slow to be useful. On the other hand, these applications often do not require an exact knowledge of the posterior. This has motivated the development of a new generation of algorithms that are fast enough to handle huge datasets but that often target an approximation of the posterior. This book gathers 18 research papers written by Approximate Bayesian Inference specialists and provides an overview of the recent advances in these algorithms. This includes optimization-based methods (such as variational approximations) and simulation-based methods (such as ABC or Monte Carlo algorithms). The theoretical aspects of Approximate Bayesian Inference are covered, specifically the PAC–Bayes bounds and regret analysis. Applications for challenging computational problems in astrophysics, finance, medical data analysis, and computer vision area also presented.

Research & information: general --- Mathematics & science --- bifurcation --- dynamical systems --- Edward–Sokal coupling --- mean-field --- Kullback–Leibler divergence --- variational inference --- Bayesian statistics --- machine learning --- variational approximations --- PAC-Bayes --- expectation-propagation --- Markov chain Monte Carlo --- Langevin Monte Carlo --- sequential Monte Carlo --- Laplace approximations --- approximate Bayesian computation --- Gibbs posterior --- MCMC --- stochastic gradients --- neural networks --- Approximate Bayesian Computation --- differential evolution --- Markov kernels --- discrete state space --- ergodicity --- Markov chain --- probably approximately correct --- variational Bayes --- Bayesian inference --- Markov Chain Monte Carlo --- Sequential Monte Carlo --- Riemann Manifold Hamiltonian Monte Carlo --- integrated nested laplace approximation --- fixed-form variational Bayes --- stochastic volatility --- network modeling --- network variability --- Stiefel manifold --- MCMC-SAEM --- data imputation --- Bethe free energy --- factor graphs --- message passing --- variational free energy --- variational message passing --- approximate Bayesian computation (ABC) --- differential privacy (DP) --- sparse vector technique (SVT) --- Gaussian --- particle flow --- variable flow --- Langevin dynamics --- Hamilton Monte Carlo --- non-reversible dynamics --- control variates --- thinning --- meta-learning --- hyperparameters --- priors --- online learning --- online optimization --- gradient descent --- statistical learning theory --- PAC–Bayes theory --- deep learning --- generalisation bounds --- Bayesian sampling --- Monte Carlo integration --- PAC-Bayes theory --- no free lunch theorems --- sequential learning --- principal curves --- data streams --- regret bounds --- greedy algorithm --- sleeping experts --- entropy --- robustness --- statistical mechanics --- complex systems