Concept# Arithmetic mean

Summary

In mathematics and statistics, the arithmetic mean (pronˌærɪθˈmɛtɪk_ˈmiːn ), arithmetic average, or just the mean or average (when the context is clear) is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results from an experiment, an observational study, or a survey. The term "arithmetic mean" is preferred in some mathematics and statistics contexts because it helps distinguish it from other types of means, such as geometric and harmonic.
In addition to mathematics and statistics, the arithmetic mean is frequently used in economics, anthropology, history, and almost every academic field to some extent. For example, per capita income is the arithmetic average income of a nation's population.
While the arithmetic mean is often used to report central tendencies, it is not a robust statistic: it is greatly influenced by outliers (values much larger or smaller than most others). For skewed distributions, such as the

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Pénélope Leyland, Fabio Nobile, Michele Pisaroni

In this work we apply the Continuation Multi-Level Monte Carlo (C-MLMC) algorithm proposed in Collier et al. (2014) to efficiently propagate operating and geometric uncertainties in inviscid compressible aerodynamics numerical simulations. The key idea of MLMC is that one can draw MC samples simultaneously and independently on several approximations of the problem under investigations on a hierarchy of nested computational grids (levels). The expectation of an output quantity is computed as a sample average using the coarsest solutions and corrected by averages of the differences of the solutions of two consecutive grids in the hierarchy. By this way, most of the computational effort is transported from the finest level (as in a standard Monte Carlo approach) to the coarsest one. The continuation algorithm (C-MLMC) is a robust and self-tuning version that estimates on the fly the optimal number of level and realizations per level. In this work we describe in detail how C-MLMC can be adapted to perform uncertainty quantification analysis in compressible aerodynamics and we apply it to the relevant test cases of a quasi 1D convergent–divergent Laval nozzle and the 2D transonic RAE-2822 airfoil.

Abelian varieties are fascinating objects, combining the fields of geometry and arithmetic. While the interest in abelian varieties has long time been of purely theoretic nature, they saw their first real-world application in cryptography in the mid 1980's, and have ever since lead to broad research on the computational and the arithmetic side. The most instructive examples of abelian varieties are elliptic curves and Jacobian varieties of hyperelliptic curves, and they come naturally equipped with some additional structure, called a principal polarization. Morphisms between abelian varieties that respect both the geometric and the arithmetic structure are called isogenies. In this thesis we focus on the computation of isogenies with cyclic kernel between principally polarized abelian varieties over finite fields.

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2010