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In this paper we prove that the stochastic Navier-Stokes equations with stable Lévy noise generate a random dynamical systems. Then we prove the existence of random attractor for the Navier-Stokes equations on 2D spheres under stable Lévy noise (finite dimensional). We also deduce the existence of a Feller Markov Invariant Measure. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Moebius measures μxn indexed by dimension n and |x| < 1 on the unit sphere Sn−1 in ℝn (n ≥ 3), and provide a precise two-sided estimate on the order of the Beckner inequality constant with exponent p ∈ [1, 2) in the three parameters. As special cases for p = 1 and p tending to 2, our results cover those in Barthe et al. [Forum Math. (submitted for publication)] for n ≥ 3 and explore an interesting phenomenon. [ABSTRACT FROM AUTHOR]

We study the extension property of isometries on the unit sphere of the l1-sum of strictly normed spaces, which is a special case of Tingley's isometric extension problem. In this paper, we will give some sufficient conditions such that such isometries can be extended to the whole space. [ABSTRACT FROM AUTHOR]

In practical situations involving mixtures formed from several ingredients, interest is sometimes centered on the response in an ellipsoidal neighborhood around a standard formulation. We show that standard, orthogonally blocked, response surface designs, defined on a q - 1 dimensional unit sphere, may be transformed into similarly orthogonally blocked q-ingredient mixture designs defined within an ellipsoid centered at the standard formulation. The method is illustrated using several examples of mixture experiments with three, four, and five ingredients, arranged in two, three, or four orthogonal blocks, obtained by projecting standard central composite designs and Box-Behnken designs into the ellipsoidal mixture region. Rotations of the resulting designs within the ellipsoidal regions are also considered. [ABSTRACT FROM AUTHOR]

We study the extension of isometries between the unit spheres of quasi-Banach spaces L p for 0 < p < 1. We give some sufficient conditions such that an isometric mapping from the the unit sphere of L p( µ) into that of another L p( ν) can be extended to be a linear isometry defined on the whole space. [ABSTRACT FROM AUTHOR]

The article reports that Navier–Stokes equations on 2D spheres coupled to thermodynamics are generally accepted as a natural model for global atmospheric circulation on Earth.