In noncommutative spaces, it is unknown whether the Pontrjagin class gives integer, as well as, the relation between the instanton number and Pontrjagin class is not clear. Where we define “Instanton number” by the size of Bα in the ADHM construction. We show the analytical derivation of the noncommuatative U(1) instanton number as an integral of Pontrjagin class (instanton charge) with the Foc...

Introduction Section 1: Preliminaries 1.1 The Navier-Stokes equations 1.2 Classical, mild and weak solutions 1.3 Navier meets Fourier Section 2: Functional setting of the equations 2.1 The Littlewood-Paley decomposition 2.2 The Besov spaces 2.3 The paraproduct rule 2.4 The wavelet decomposition 2.5 Other useful function spaces Section 3: Existence theorems 3.1 The fixed point theorem 3.2 Scalin...

In this note, we consider the homogenization of compressible Navier-Stokes equations in a periodically perforated domain $\mathbb{R}^3$. Assuming that particle size scales like $\varepsilon^3$, where $\varepsilon>0$ is their mutual distance, and Mach number decreases fast enough, show limit $\varepsilon\to 0$, velocity density converge to solution incompressible with Brinkman term. We strongly ...

Simple eigenvalue tests are given to ascertain that a given real 4X 4 matrix transforms the four-vector of Stokes parameters of a beam of light into the four-vector of Stokes parameters of another beam of light, and to determine whether a given 4X4 matrix is a weighted sum of pure Mueller matrices. The latter result is derived for matrices satisfying a certain symmetry condition. To derive thes...

In noncommutative spaces it is unknown whether the Pontrjagin class gives integer, as well as, the relation between the instanton number and Pontrjagin class is not clear. Where we define “Instanton number” by the size of Bα in the ADHM construction. We show the analytical derivation of the noncommuatative U(1) instanton number as an integral of Pontrjagin class (instanton charge) in the Fock s...

Spatial and velocity statistics of heavy point-like particles in incompressible, homogeneous, and isotropic three-dimensional turbulence is studied by means of direct numerical simulations at two values of the Taylor-scale Reynolds number Reλ ∼ 200 and Reλ ∼ 400, corresponding to resolutions of 512 and 2048 grid points, respectively. Particles Stokes number values range from St ≈ 0.2 to 70. Sta...

Local preconditioning for the Navier-Stokes equations may be called optimal if it equalizes all propagation and dissipation time-scales, for all combinations of Mach number and Reynolds number. Previously designed preconditioners are ineffective for certain combinations of low Reynolds number and low Mach number; in addition some of these create a growing mode, making the PDE-system unstable. (...