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This Special Issue brings together original research papers, in all areas of mathematics, that are concerned with inequalities or the role of inequalities. The research results presented in this Special Issue are related to improvements in classical inequalities, highlighting their applications and promoting an exchange of ideas between mathematicians from many parts of the world dedicated to the theory of inequalities. This volume will be of interest to mathematicians specializing in inequality theory and beyond. Many of the studies presented here can be very useful in demonstrating new results. It is our great pleasure to publish this book. All contents were peer-reviewed by multiple referees and published as papers in our Special Issue in the journal Symmetry. These studies give new and interesting results in mathematical inequalities enabling readers to obtain the latest developments in the fields of mathematical inequalities. Finally, we would like to thank all the authors who have published their valuable work in this Special Issue. We would also like to thank the editors of the journal Symmetry for their help in making this volume, especially Mrs. Teresa Yu.

Research & information: general --- Geography --- Ostrowski inequality --- Hölder's inequality --- power mean integral inequality --- n-polynomial exponentially s-convex function --- weight coefficient --- Euler-Maclaurin summation formula --- Abel's partial summation formula --- half-discrete Hilbert-type inequality --- upper limit function --- Hermite-Hadamard inequality --- (p, q)-calculus --- convex functions --- trapezoid-type inequality --- fractional integrals --- functions of bounded variations --- (p,q)-integral --- post quantum calculus --- convex function --- a priori bounds --- 2D primitive equations --- continuous dependence --- heat source --- Jensen functional --- A-G-H inequalities --- global bounds --- power means --- Simpson-type inequalities --- thermoelastic plate --- Phragmén-Lindelöf alternative --- Saint-Venant principle --- biharmonic equation --- symmetric function --- Schur-convexity --- inequality --- special means --- Shannon entropy --- Tsallis entropy --- Fermi-Dirac entropy --- Bose-Einstein entropy --- arithmetic mean --- geometric mean --- Young's inequality --- Simpson's inequalities --- post-quantum calculus --- spatial decay estimates --- Brinkman equations --- midpoint and trapezoidal inequality --- Simpson's inequality --- harmonically convex functions --- Simpson inequality --- (n,m)-generalized convexity --- Ostrowski inequality --- Hölder's inequality --- power mean integral inequality --- n-polynomial exponentially s-convex function --- weight coefficient --- Euler-Maclaurin summation formula --- Abel's partial summation formula --- half-discrete Hilbert-type inequality --- upper limit function --- Hermite-Hadamard inequality --- (p, q)-calculus --- convex functions --- trapezoid-type inequality --- fractional integrals --- functions of bounded variations --- (p,q)-integral --- post quantum calculus --- convex function --- a priori bounds --- 2D primitive equations --- continuous dependence --- heat source --- Jensen functional --- A-G-H inequalities --- global bounds --- power means --- Simpson-type inequalities --- thermoelastic plate --- Phragmén-Lindelöf alternative --- Saint-Venant principle --- biharmonic equation --- symmetric function --- Schur-convexity --- inequality --- special means --- Shannon entropy --- Tsallis entropy --- Fermi-Dirac entropy --- Bose-Einstein entropy --- arithmetic mean --- geometric mean --- Young's inequality --- Simpson's inequalities --- post-quantum calculus --- spatial decay estimates --- Brinkman equations --- midpoint and trapezoidal inequality --- Simpson's inequality --- harmonically convex functions --- Simpson inequality --- (n,m)-generalized convexity

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This book describes the latest advances in the theory of mean field games, which are optimal control problems with a continuum of players, each of them interacting with the whole statistical distribution of a population. While originating in economics, this theory now has applications in areas as diverse as mathematical finance, crowd phenomena, epidemiology, and cybersecurity.Because mean field games concern the interactions of infinitely many players in an optimal control framework, one expects them to appear as the limit for Nash equilibria of differential games with finitely many players, as the number of players tends to infinity. This book rigorously establishes this convergence, which has been an open problem until now. The limit of the system associated with differential games with finitely many players is described by the so-called master equation, a nonlocal transport equation in the space of measures. After defining a suitable notion of differentiability in the space of measures, the authors provide a complete self-contained analysis of the master equation. Their analysis includes the case of common noise problems in which all the players are affected by a common Brownian motion. They then go on to explain how to use the master equation to prove the mean field limit.This groundbreaking book presents two important new results in mean field games that contribute to a unified theoretical framework for this exciting and fast-developing area of mathematics.

Convergence. --- Mean field theory. --- Many-body problem --- Statistical mechanics --- Functions --- A priori estimate. --- Approximation. --- Bellman equation. --- Boltzmann equation. --- Boundary value problem. --- C0. --- Chain rule. --- Compact space. --- Computation. --- Conditional probability distribution. --- Continuous function. --- Convergence problem. --- Convex set. --- Cooperative game. --- Corollary. --- Decision-making. --- Derivative. --- Deterministic system. --- Differentiable function. --- Directional derivative. --- Discrete time and continuous time. --- Discretization. --- Dynamic programming. --- Emergence. --- Empirical distribution function. --- Equation. --- Estimation. --- Euclidean space. --- Folk theorem (game theory). --- Folk theorem. --- Heat equation. --- Hermitian adjoint. --- Implementation. --- Initial condition. --- Integer. --- Large numbers. --- Linearization. --- Lipschitz continuity. --- Lp space. --- Macroeconomic model. --- Markov process. --- Martingale (probability theory). --- Master equation. --- Mathematical optimization. --- Maximum principle. --- Method of characteristics. --- Metric space. --- Monograph. --- Monotonic function. --- Nash equilibrium. --- Neumann boundary condition. --- Nonlinear system. --- Notation. --- Numerical analysis. --- Optimal control. --- Parameter. --- Partial differential equation. --- Periodic boundary conditions. --- Porous medium. --- Probability measure. --- Probability theory. --- Probability. --- Random function. --- Random variable. --- Randomization. --- Rate of convergence. --- Regime. --- Scientific notation. --- Semigroup. --- Simultaneous equations. --- Small number. --- Smoothness. --- Space form. --- State space. --- State variable. --- Stochastic calculus. --- Stochastic control. --- Stochastic process. --- Stochastic. --- Subset. --- Suggestion. --- Symmetric function. --- Technology. --- Theorem. --- Theory. --- Time consistency. --- Time derivative. --- Uniqueness. --- Variable (mathematics). --- Vector space. --- Viscosity solution. --- Wasserstein metric. --- Weak solution. --- Wiener process. --- Without loss of generality.