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This book deals with the reliable verification of the accuracy of approximate solutions which is one of the central problems in modern applied analysis.After giving an overview of the methods developed for models based on partial differential equations, the author derives computable a posteriori error estimates by using methods of the theory of partial differential equations and functional analysis. These estimates are applicable to approximate solutions computed by various methods.

Differential equations, Partial. --- Error analysis (Mathematics) --- Errors, Theory of --- Instrumental variables (Statistics) --- Mathematical statistics --- Numerical analysis --- Statistics --- Partial differential equations --- A-posteriori-Abschätzung. --- Error analysis (Mathematics). --- Partielle Differentialgleichung. --- Partial Differential Equation, Approximate Solution, A Posteriori Error Estimates.

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Engineering practice requires the use of structures containing identical components or parts, useful from several points of view: less information is needed to describe the system, design is made quicker and easier, components are made faster than a complex assembly, and finally the time to achieve the structure and the cost of manufacturing decreases. Additionally, the subsequent maintenance of the system becomes easier and cheaper. This Special Issue is dedicated to this kind of mechanical structure, describing the properties and methods of analysis of these structures. Discrete or continuous structures in static and dynamic cases are considered. Theoretical models, mathematical methods, and numerical analyses of the systems, such as the finite element method and experimental methods, are expected to be used in the research. Machine building, automotive, aerospace, and civil engineering are the main areas in which such applications appear, but they are found in most engineering fields.

History of engineering & technology --- exact solutions --- the generalized Benjamin–Bona–Mahony equation --- generalized exponential rational function method --- solitary wave solutions --- symbolic computation --- Thomson effect --- initial stress --- magneto-thermoelastic --- voids --- normal mode method --- G-N theory --- eigenvalue problem --- axisymmetric and non-axisymmetric vibrations --- multiparametric special functions --- circular plate --- functionally graded porous material --- initially stressed bodies --- dipolar structure --- volume fraction --- domain of influence --- chaotic time series prediction --- neural network --- firefly algorithm --- CEEMDAN --- VMD --- symmetric geometry --- guitar’s plate --- modal analysis --- skew symmetric eigenmodes --- comparison --- Friedman test --- linear regression --- nonlinear regression --- sign test --- symmetric errors --- Wilcoxon test --- coefficient of variation --- ratio --- symmetric and asymmetric distributions --- test of hypothesis --- multi-body system --- finite element method (FEM) --- linear elastic elements --- Lagrange’s equations --- two-dimensional finite element --- plane motion --- backward in time problem --- dipolar thermoelastic body --- uniqueness of solution --- Cesaro means --- partition of energies --- α-fractional calculus --- vibration isolation --- fractional-order differential equation --- rubber-like elastomers --- Riemann–Liouville/Caputo/Grünwald-Letnikov fractional derivative --- advanced mechanics --- analytical dynamics --- acceleration energies --- robotics --- viscoelasticity --- type III thermal law --- finite elements --- error estimates --- numerical results --- symmetry --- topology --- mechanical structures --- robots --- vibration --- mechanical engineering --- applied mechanics --- bird-strike impact --- composite laminate --- damage equivalency --- fan blade --- symmetrical configurations

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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. --- 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.