The surjectivity of the operator $D_2+i{x}_2^{2k}{D}_1$ from the Gevrey space ${\mathcal{E}}^{\left\{s\right\}}\left({\mathbb{R}}^2\right)$, $s\ge 1$, onto itself and its non-surjectivity from ${\mathcal{E}}^{\left\{s\right\}}\left({\mathbb{R}}^3\right)$ to ${\mathcal{E}}^{\left\{s\right\}}\left({\mathbb{R}}^3\right)$ is proved.

Let: $𝐘=\left({𝐘}_i\right)$, where ${𝐘}_i=\left(Y_{i,1},...,Y_{i,d}\right)$, $i=1,2,\cdots$, be a $d$-dimensional, identically distributed, stationary, centered process with uniform marginals and a joint cdf $F$, and $F_n\left(𝐱\right):=\frac{1}{n}{\sum }_{i=1}^n𝕀\left(Y_{i,1}\le x_1,\cdots ,Y_{i,d}\le x_d\right)$ denote the corresponding empirical cdf. In our work, we prove the almost sure central limit theorem for an empirical process $B_n=\sqrt{n}\left(F_n-F\right)$ under some weak dependence conditions due to Doukhan and Louhichi. Some application of the established result to copula processes is also presented.

Let $\mathrm{\Omega }$ be a bounded open convex set of class $C^2$. Let $a\left(x,H\left(u\right)\right)$ be a non linear operator satisfying the condition (A) (elliptic) with constants $\alpha$, $\gamma$, $\delta$. We prove that a number $\lambda \ge 0$ is an eigenvalue for the operator $a\left(x,H\left(u\right)\right)$ if and only if the number $\alpha \lambda$ is an eigen-value for the operator $\mathrm{\Delta }u$. If $\lambda \ge 0$ , the two systems $a\left(x,H\left(u\right)\right)=\lambda u$ and $\mathrm{\Delta }u=\alpha \lambda u$ have the same solutions. In particular, also the eventual eigen-values of the operator $a\left(x,H\left(u\right)\right)$ should all be negative. Finally, we obtain a sufficient condition for the existence of solutions $u\in H^2\cap H_0^1\left(\mathrm{\Omega }\right)$...

Let [...] f(z)=∑n=0∞αnzn $f\left(z\right)={\sum }_{n=0}^{\infty }{\alpha }_n{z}^n$ be a function defined by power series with complex coefficients and convergent on the open disk D (0, R) ⊂ ℂ, R > 0. For any x, y ∈ ℬ, a Banach algebra, with ‖x‖, ‖y‖ < R we show among others that [...] ‖f(y)−f(x)‖≤‖y−x‖∫01fa′(‖(1−t)x+ty‖)dt $$∥f\left(y\right)-f\left(x\right)∥\le ∥y-x∥{\int }_0^1{f}_a^{\text{'}}\left(∥(1-t)x+ty∥\right)dt$$ where [...] fa(z)=∑n=0∞|αn| zn $f_a\left(z\right)={\sum }_{n=0}^{\infty }|{\alpha }_n|z^n$ . Inequalities for the commutator such as [...] ‖f(x)f(y)−f(y)f(x)‖≤2fa(M)fa′(M)‖y−x‖, $$∥f\left(x\right)f\left(y\right)-f\left(y\right)f\left(x\right)∥\le 2f_a\left(M\right)f_a^{\text{'}}\left(M\right)∥y-x∥,$$ if ‖x‖, ‖y‖ ≤ M < R, as well as some inequalities of Hermite–Hadamard type are also provided. ...

We list some open problems concerning the polarized partition relation. We solve a couple of them, by showing that for every limit non-inaccessible ordinal α there exists a forcing notion ℙ such that the strong polarized relation $\left(\genfrac{}{}{0pt}{}{{\aleph }_{\alpha +1}}{{\aleph }_{\alpha }}\right)\to {\left(\genfrac{}{}{0pt}{}{{\aleph }_{\alpha +1}}{{\aleph }_{\alpha }}\right)}_2^{1,1}$ holds in $V^{\mathbb{P}}$.

$\mathrm{\Omega }$ is a bounded open set of ${\mathbb{R}}^n$, of class $C^2$ and $T>0$. In the cylinder $Q=\mathrm{\Omega }\times \left(0,T\right)$ we consider non variational basic operator $a\left(H\left(u\right)\right)-\partial u/\partial t$ where $a\left(\xi \right)$ is a vector in ${\mathbb{R}}^N$, $N\ge 1$, which is continuous in $\xi$ and satisfies the condition (A). It is shown that $\mathrm{\forall }f\in L^2\left(Q\right)$ the Cauchy-Dirichlet problem $u\in W_0^{2,1}\left(Q\right)$, $a\left(H\left(u\right)\right)-\partial u/\partial t=f$ in $Q$, has a unique solution. It is further shown that if $u\in W_0^{2,1}\left(Q\right)$ is a solution of the basic system $a\left(H\left(u\right)\right)-\partial u/\partial t=0$ in $Q$, then $H\left(u\right)$ and $\partial u/\partial t$ belong to $H_{loc}^1\left(Q\right)$. From this the Hölder continuity in $Q$ of the vectors $u$ and $Du$ are deduced respectively when $n\le 4$ and...