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Concept

Haar measure

Summary

In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This measure was introduced by Alfréd Haar in 1933, though its special case for Lie groups had been introduced by Adolf Hurwitz in 1897 under the name "invariant integral". Haar measures are used in many parts of analysis, number theory, group theory, representation theory, statistics, probability theory, and ergodic theory. Preliminaries Let (G, \cdot) be a locally compact Hausdorff topological group. The \sigma-algebra generated by all open subsets of G is called the Borel algebra. An element of the Borel algebra is called a Borel set. If g is an element of G and S is a subset of G, then we define the left and right translates of S by g as follows:

Left translate:

Official source

https://en.wikipedia.org/wiki/Haar_measure

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Invariant Measures of Gaussian Type for 2D Turbulence

Gaussian measures μβ,ν are associated to some stochastic 2D models of turbulence.They are Gibbs measures constructed by means of an invariant quantity of the system depending on some parameter β (related to the 2D nature of the fluid) and the viscosity ν.We prove the existence and the uniqueness of the global flow for the stochastic viscous system; moreover the measure μβ,ν is invariant for this flow and is the unique invariant measure. Finally, we prove that the deterministic inviscid equation has a μβ,ν-stationary solution (for any ν >0).

Springer2012

In this thesis we compute motivic classes of hypertoric varieties, Nakajima quiver varieties and open de Rham spaces in a certain localization of the Grothendieck ring of varieties. Furthermore we study the

p

-adic pushforward of the Haar measure under a hypertoric moment map

\mu

. This leads to an explicit formula for the Igusa zeta function

\FI_\mu(s)

\mu

, and in particular to a small set of candidate poles for

\FI_\mu(s)

. We also study various properties of the residue at the largest pole of

\FI_\mu(s)

. Finally, if

\mu

is constructed out of a quiver

\Gamma

we give a conjectural description of this residue in terms of indecomposable representations of

\Gamma

over finite depth rings. The connections between these different results is the method of proof. At the heart of each theorem lies a motivic or

p

-adic volume computation, which is only possible due to some surprising cancellations. These cancellations are reminiscent of a result in classical symplectic geometry by Duistermaat and Heckman on the localization of the Liouville measure, hence the title of the thesis.

EPFL2017

A Bayesian numerical homogenization method for elliptic multiscale inverse problems

Assyr Abdulle, Andrea Di Blasio

A new strategy based on numerical homogenization and Bayesian techniques for solving multiscale inverse problems is introduced. We consider a class of elliptic problems which vary at a microscopic scale, and we aim at recovering the highly oscillatory tensor from measurements of the fine scale solution at the boundary, using a coarse model based on numerical homogenization and model order reduction. We provide a rigorous Bayesian formulation of the problem, taking into account different possibilities for the choice of the prior measure. We prove well-posedness of the effective posterior measure and, by means of G-convergence, we establish a link between the effective posterior and the fine scale model. Several numerical experiments illustrate the efficiency of the proposed scheme and confirm the theoretical findings.

2020

p