This paper is devoted to the study of a turbulent circulation model. Equations are derived from the “Navier-Stokes turbulent kinetic energy” system. Some simplifications are performed but attention is focused on non linearities linked to turbulent eddy viscosity ${\nu }_t$. The mixing length $\ell$ acts as a parameter which controls the turbulent part in ${\nu }_t$. The main theoretical results that we have obtained concern the uniqueness of the solution for bounded eddy viscosities and small values of $\ell$...

In this paper, we consider a Borel measurable function on the space of $m\times n$ matrices $f:M^{m\times n}\to \overline{ℝ}$ taking the value $+\infty$, such that its rank-one-convex envelope $Rf$ is finite and satisfies for some fixed $p>1$: $$-c_0\le Rf\left(F\right)\le {c(1+\parallel F\parallel }^p)\text{for}\text{all}F\in M^{m\times n},$$ where $c,c_0>0$. Let $Ø$ be a given regular bounded open domain of ${ℝ}^n$. We define on $W^{1,p}(Ø;{ℝ}^m)$ the functional $$I\left(u\right)={\int }_{Ø}f\left(\nabla u\left(x\right)\right)dx.$$ Then, under some technical restrictions on $f$, we show that the relaxed functional $\overline{I}$ for the weak topology of $W^{1,p}(Ø;{ℝ}^m)$ has the integral representation: $$\overline{I}\left(u\right)={\int }_{Ø}Q\left[Rf\right]\left(\nabla u\left(x\right)\right)dx,$$ where for a given function $g$, $Qg$ denotes...

A Large Deviation Principle (LDP) is proved for the family $\frac{1}{n}{\sum }_1^n𝐟\left(x_i^n\right)·Z_i^n$ where the deterministic probability measure $\frac{1}{n}{\sum }_1^n{\delta }_{x_i^n}$ converges weakly to a probability measure R and ${\left(Z_i^n\right)}_{i\in \mathbb{N}}$ are ${ℝ}^d$-valued independent random variables whose distribution depends on $x_i^n$ and satisfies the following exponential moments condition: $$\underset{i,n}{sup}𝔼{\mathrm{e}}^{{\alpha }^{*}\left|Z_i^n\right|}<+\infty \mathrm{forsome}0<{\alpha }^{*}<+\infty .$$ In this context, the identification of the rate function is non-trivial due to the absence of equidistribution. We rely on fine convex analysis...

The complexity of computing, via threshold circuits, the and of fixed-dimension $k\times k$ matrices with integer or rational entries is studied. We call these two problems ${\mathrm{𝖨𝖬𝖯}}_{𝗄}$ and ${\mathrm{𝖬𝖯𝖮𝖶}}_{𝗄}$, respectively, for short. We prove that: (i) For $k\ge 2$, ${\mathrm{𝖨𝖬𝖯}}_{𝗄}$ does not belong to ${\mathrm{TC}}^0$, unless ${\mathrm{TC}}^0={\mathrm{NC}}^1$.newline (ii) For : ${\mathrm{𝖨𝖬𝖯}}_{\mathsf{2}}$ belongs to ${\mathrm{TC}}^0$ while, for $k\ge 3$, ${\mathrm{𝖨𝖬𝖯}}_{𝗄}$ does not belong to ${\mathrm{TC}}^0$, unless ${\mathrm{TC}}^0={\mathrm{NC}}^1$. (iii) For any , ${\mathrm{𝖬𝖯𝖮𝖶}}_{𝗄}$ belongs to ${\mathrm{TC}}^0$.