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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.

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This text provides a definitive proof of global nonlinear stability of Minkowski space-time as a solution of the Einstein-Klein-Gordon equations of general relativity. Along the way, a novel robust analytical framework is developed, which extends to more general matter models.

General relativity (Physics). --- Klein-Gordon equation. --- Mathematical physics. --- Quantum field theory. --- SCIENCE / Physics / Mathematical & Computational. --- Addition. --- Algebraic structure. --- Antiderivative. --- Approximation. --- Asymptote. --- Asymptotic analysis. --- Bending. --- Big O notation. --- Bootstrapping (statistics). --- Calculation. --- Cauchy distribution. --- Coefficient. --- Combination. --- Compact space. --- Complex number. --- Computation. --- Conserved quantity. --- Coordinate system. --- Coordinate-free. --- Covariant derivative. --- Derivative. --- Differential operator. --- Dispersion relation. --- Einstein field equations. --- Energy functional. --- Equation. --- Estimation. --- Exponential growth. --- Foliation. --- Fourier analysis. --- Fourier transform. --- Function (mathematics). --- Function space. --- General relativity. --- Geodesic. --- Geodesics in general relativity. --- Geographic coordinate system. --- Geometry. --- Global analysis. --- Globality. --- High frequency. --- Hyperboloid. --- Hypersurface. --- Hypothesis. --- Implementation. --- Ingredient. --- Integration by parts. --- Interpolation inequality. --- Klein–Gordon equation. --- Light cone. --- Local coordinates. --- Mathematical optimization. --- Metric tensor (general relativity). --- Metric tensor. --- Minkowski space. --- Momentum. --- Monograph. --- Monotonic function. --- Nonlinear system. --- Optics. --- Parametrization. --- Partial differential equation. --- Pointwise. --- Poisson bracket. --- Quantity. --- Remainder. --- Result. --- Riemann curvature tensor. --- Scalar field. --- Scattering. --- Schwarzschild metric. --- Scientific notation. --- Second fundamental form. --- Simultaneous equations. --- Small data. --- Small number. --- Sobolev space. --- Soliton. --- Space. --- Stability theory. --- Stress–energy tensor. --- Support (mathematics). --- Symmetrization. --- Theorem. --- Time derivative. --- Timelike Infinity. --- Trace (linear algebra). --- Two-dimensional space. --- Vacuum. --- Vector field. --- Very low frequency. --- Relativistic quantum field theory --- Field theory (Physics) --- Quantum theory --- Relativity (Physics) --- Physical mathematics --- Physics --- Schrödinger-Klein-Gordon equation --- Quantum field theory --- Wave equation --- Relativistic theory of gravitation --- Relativity theory, General --- Gravitation --- Mathematics --- General relativity (Physics) --- Science. --- Physics.

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Benford's law states that the leading digits of many data sets are not uniformly distributed from one through nine, but rather exhibit a profound bias. This bias is evident in everything from electricity bills and street addresses to stock prices, population numbers, mortality rates, and the lengths of rivers. Here, Steven Miller brings together many of the world's leading experts on Benford's law to demonstrate the many useful techniques that arise from the law, show how truly multidisciplinary it is, and encourage collaboration.Beginning with the general theory, the contributors explain the prevalence of the bias, highlighting explanations for when systems should and should not follow Benford's law and how quickly such behavior sets in. They go on to discuss important applications in disciplines ranging from accounting and economics to psychology and the natural sciences. The contributors describe how Benford's law has been successfully used to expose fraud in elections, medical tests, tax filings, and financial reports. Additionally, numerous problems, background materials, and technical details are available online to help instructors create courses around the book.Emphasizing common challenges and techniques across the disciplines, this accessible book shows how Benford's law can serve as a productive meeting ground for researchers and practitioners in diverse fields.

Distribution (Probability theory) --- Probability measures. --- 1938 paper. --- 2009 presidential elections. --- Benford analysis. --- Benford distribution. --- Benford property. --- Benford test. --- Benford's law geometry. --- Benford's law limit. --- Benford's law. --- Benford-good system. --- Eric Poehlman. --- European statistics. --- FSDs. --- Fourier analysis. --- Frank Benford. --- Fundamental Equivalence. --- Greek statistics. --- Iranian election. --- Iranian presidential elections. --- Lvy processes. --- PV effect. --- Partial Volume effect. --- Poisson Summation Formula. --- Poisson Summation. --- Simon Newcomb. --- Standard Condition. --- VAR. --- Value at Risk. --- accounting programs. --- accounting students. --- accounting. --- auditing. --- authentic data sets. --- behavioral approaches. --- bias. --- complaint data sets. --- computer systems. --- cumulative distribution. --- data sets. --- data-adaptive methods. --- decision-making research. --- densities. --- deterministic processes. --- digit bias. --- direct applications. --- econometric regression. --- economics. --- election fraud. --- elections. --- empirical economics research. --- empirical economics. --- error detection. --- explicit error bounds. --- explicit error estimates. --- exponential Lvy processes. --- finance. --- financial reports. --- financial statistics. --- first significant digits. --- first-digit analysis. --- first-digit frequency. --- fixed odds. --- forecasts. --- fraud detection. --- fraud. --- fraudulent data sets. --- functions. --- gambling. --- generic potential applications. --- geometry. --- government deficit. --- information-theoretic methods. --- local boostrap model. --- logarithms. --- lottery. --- macroeconomic data. --- managing risk. --- mathematical theory. --- meaningful numbers. --- medical tests. --- misreporting. --- natural data. --- natural sciences. --- non-uniformity. --- normalized functionals. --- number lottery games. --- numbers games. --- origins. --- parametric distributions. --- probability distributions. --- psychology. --- random processes. --- replication. --- scale invariance. --- scientific data sets. --- scientific misconduct. --- second digits. --- significand. --- significant digits. --- small number. --- social statistics. --- statistical relationship. --- statistics education. --- statistics. --- tampering. --- tax filing. --- tax fraud. --- total variation. --- uniform distribution. --- vote counts. --- voting.