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The work described in this book originates from a major effort to develop a fundamental theory of the glass and the jamming transitions. The first chapters guide the reader through the phenomenology of supercooled liquids and structural glasses and provide the tools to analyze the most frequently used models able to predict the complex behavior of such systems. A fundamental outcome is a detailed theoretical derivation of an effective thermodynamic potential, along with the study of anomalous vibrational properties of sphere systems. The interested reader can find in these pages a clear and deep analysis of mean-field models as well as the description of advanced beyond-mean-field perturbative expansions. To investigate important second-order phase transitions in lattice models, the last part of the book proposes an innovative theoretical approach, based on a multi-layer construction. The different methods developed in this thesis shed new light on important connections among constraint satisfaction problems, jamming and critical phenomena in complex systems, and lay part of the groundwork for a complete theory of amorphous solids.

Phase transformations (Statistical physics) --- Mean field theory. --- Phase Transitions and Multiphase Systems. --- Ceramics, Glass, Composites, Natural Materials. --- Low Temperature Physics. --- Phase transitions (Statistical physics). --- Ceramics. --- Glass. --- Composites (Materials). --- Composite materials. --- Low temperature physics. --- Low temperatures. --- Cryogenics --- Low temperature physics --- Temperatures, Low --- Temperature --- Cold --- Composites (Materials) --- Multiphase materials --- Reinforced solids --- Solids, Reinforced --- Two phase materials --- Materials --- Amorphous substances --- Ceramics --- Glazing --- Ceramic technology --- Industrial ceramics --- Keramics --- Building materials --- Chemistry, Technical --- Clay --- Phase changes (Statistical physics) --- Phase transitions (Statistical physics) --- Phase rule and equilibrium --- Statistical physics

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This book covers the theory of electronic structure of materials, with special emphasis on the usage of linear muffin-tin orbitals. Methodological aspects are given in detail as are examples of the method when applied to various materials. Different exchange and correlation functionals are described and how they are implemented within the basis of linear muffin-tin orbitals. Functionals covered are the local spin density approximation, generalised gradient approximation, self-interaction correction and dynamical mean field theory.

Density functionals. --- Electronic structure. --- Electronics. --- Mean field theory. --- Physics --- Physical Sciences & Mathematics --- Atomic Physics --- Electronic structure --- Mathematical physics. --- Materials. --- Mathematical models. --- Engineering --- Engineering materials --- Industrial materials --- Physical mathematics --- Density functional methods --- Density functional theory --- Functional methods, Density --- Functionals, Density --- Structure, Electronic --- Materials --- Mathematics --- Physics. --- Condensed matter. --- Condensed Matter Physics. --- Mathematical Methods in Physics. --- Engineering design --- Manufacturing processes --- Functional analysis --- Atomic structure --- Energy-band theory of solids --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Condensed materials --- Condensed media --- Condensed phase --- Materials, Condensed --- Media, Condensed --- Phase, Condensed --- Liquids --- Matter --- Solids

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This two-volume book offers a comprehensive treatment of the probabilistic approach to mean field game models and their applications. The book is self-contained in nature and includes original material and applications with explicit examples throughout, including numerical solutions. Volume II tackles the analysis of mean field games in which the players are affected by a common source of noise. The first part of the volume introduces and studies the concepts of weak and strong equilibria, and establishes general solvability results. The second part is devoted to the study of the master equation, a partial differential equation satisfied by the value function of the game over the space of probability measures. Existence of viscosity and classical solutions are proven and used to study asymptotics of games with finitely many players. Together, both Volume I and Volume II will greatly benefit mathematical graduate students and researchers interested in mean field games. The authors provide a detailed road map through the book allowing different access points for different readers and building up the level of technical detail. The accessible approach and overview will allow interested researchers in the applied sciences to obtain a clear overview of the state of the art in mean field games.

Mean field theory. --- Game theory. --- Mathematics. --- Partial differential equations. --- Calculus of variations. --- Probabilities. --- Economic theory. --- Probability Theory and Stochastic Processes. --- Calculus of Variations and Optimal Control; Optimization. --- Partial Differential Equations. --- Economic Theory/Quantitative Economics/Mathematical Methods. --- Many-body problem --- Statistical mechanics --- Games, Theory of --- Theory of games --- Mathematical models --- Mathematics --- Distribution (Probability theory. --- Mathematical optimization. --- Differential equations, partial. --- Economic theory --- Political economy --- Social sciences --- Economic man --- Partial differential equations --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Mathematical analysis --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Isoperimetrical problems --- Variations, Calculus of --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk

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Flows of thermal origin and heat transfer problems are central in a variety of disciplines and industrial applications. The present book entitled Thermal Flows consists of a collection of studies by distinct investigators and research groups dealing with different types of flows relevant to both natural and technological contexts. Both reviews of the state-of-the-art and new theoretical, numerical and experimental investigations are presented, which illustrate the structure of these flows, their stability behavior, and the possible bifurcations to different patterns of symmetry and/or spatiotemporal regimes. Moreover, different categories of fluids are considered (liquid metals, gases, common fluids such as water and silicone oils, organic and inorganic transparent liquids, and nanofluids). This information is presented under the hope that it will serve as a new important resource for physicists, engineers and advanced students interested in the physics of non-isothermal fluid systems; fluid mechanics; environmental phenomena; meteorology; geophysics; and thermal, mechanical and materials engineering.

Research & information: general --- Physics --- coating flow --- free surface --- boundary layer --- stress singularity --- matched asymptotic expansions --- computational fluid dynamics --- turbulence --- rotating thermal convection --- Rayleigh-Bénard --- heat enhancement --- nanofluid --- circular pipe --- twisted tape --- porous media --- metal foam --- convection-driven dynamos --- numerical simulations --- bistability --- mean-field magnetohydrodynamics --- spherical shells --- stochastic equations --- equivalence of measures --- nature of turbulence --- critical Reynolds number --- thermovibrational convection --- gravity modulation --- thermofluid-dynamic distortions --- patterning behavior --- stratified mixing layer --- non-modal instability --- Kelvin-Helmholtz instability --- Holmboe instability --- rotating thermal magnetoconvection --- linear onset --- sphere --- Rayleigh-Bénard convection --- time periodical cooling --- Lattice Boltzmann method --- thermocapillary-driven convection --- half-zone liquid bridges --- particles --- coherent structures --- particle accumulation structure (PAS) --- high Prandtl number fluids --- plane layer --- circular translational vibrations --- thermal vibrational convection --- convective patterns --- coating flow --- free surface --- boundary layer --- stress singularity --- matched asymptotic expansions --- computational fluid dynamics --- turbulence --- rotating thermal convection --- Rayleigh-Bénard --- heat enhancement --- nanofluid --- circular pipe --- twisted tape --- porous media --- metal foam --- convection-driven dynamos --- numerical simulations --- bistability --- mean-field magnetohydrodynamics --- spherical shells --- stochastic equations --- equivalence of measures --- nature of turbulence --- critical Reynolds number --- thermovibrational convection --- gravity modulation --- thermofluid-dynamic distortions --- patterning behavior --- stratified mixing layer --- non-modal instability --- Kelvin-Helmholtz instability --- Holmboe instability --- rotating thermal magnetoconvection --- linear onset --- sphere --- Rayleigh-Bénard convection --- time periodical cooling --- Lattice Boltzmann method --- thermocapillary-driven convection --- half-zone liquid bridges --- particles --- coherent structures --- particle accumulation structure (PAS) --- high Prandtl number fluids --- plane layer --- circular translational vibrations --- thermal vibrational convection --- convective patterns

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Mean field approximation has been adopted to describe macroscopic phenomena from microscopic overviews. It is still in progress; fluid mechanics, gauge theory, plasma physics, quantum chemistry, mathematical oncology, non-equilibirum thermodynamics.  spite of such a wide range of scientific areas that are concerned with the mean field theory, a unified study of its mathematical structure has not been discussed explicitly in the open literature.  The benefit of this point of view on nonlinear problems should have significant impact on future research, as will be seen from the underlying features of self-assembly or bottom-up self-organization which is to be illustrated in a unified way. The aim of this book is to formulate the variational and hierarchical aspects of the equations that arise in the mean field theory from macroscopic profiles to microscopic principles, from dynamics to equilibrium, and from biological models to models that arise from chemistry and physics.

Applied Mathematics --- Engineering & Applied Sciences --- Mean field theory. --- Differential equations, Partial. --- Partial differential equations --- Mathematics. --- Mathematical analysis. --- Analysis (Mathematics). --- Calculus of variations. --- Biomathematics. --- Mathematical physics. --- Analysis. --- Calculus of Variations and Optimal Control; Optimization. --- Mathematical Physics. --- Genetics and Population Dynamics. --- Physiological, Cellular and Medical Topics. --- Many-body problem --- Statistical mechanics --- Global analysis (Mathematics). --- Mathematical optimization. --- Genetics --- Physiology --- Animal physiology --- Animals --- Biology --- Anatomy --- Embryology --- Mendel's law --- Adaptation (Biology) --- Breeding --- Chromosomes --- Heredity --- Mutation (Biology) --- Variation (Biology) --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Mathematical analysis --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Mathematics --- Physical mathematics --- Physics --- Isoperimetrical problems --- Variations, Calculus of --- 517.1 Mathematical analysis