We consider a function $u:\Omega \to {ℝ}^N$, $\Omega \subset {ℝ}^n$, minimizing the integral ${\int }_{\Omega }\left(\right|D_1{u|}^2+\cdots +|D_{n-1}{u|}^2+|D_{n}u{|}^p)dx$, $2(n+1)/(n+3)\le p<2$, where $D_{i}u=\partial u/\partial x_i$, or some more general functional with the same behaviour; we prove the existence of second weak derivatives $D\left(D_{1}u\right),\cdots ,D\left(D_{n-1}u\right)\in L^2$ and $D\left(D_{n}u\right)\in L^p$.

$\alpha$-normality and $\beta$-normality are properties generalizing normality of topological spaces. They consist in separating dense subsets of closed disjoint sets. We construct an example of a Tychonoff $\beta$-normal non-normal space and an example of a Hausdorff $\alpha$-normal non-regular space.

Let $X$ and $Y$ be vector spaces over the same field $K$. Following the terminology of Richard Arens [Pacific J. Math. 11 (1961), 9–23], a relation $F$ of $X$ into $Y$ is called linear if $\lambda F\left(x\right)\subset F\left(\lambda x\right)$ and $F\left(x\right)+F\left(y\right)\subset F(x+y)$ for all $\lambda \in K\setminus \left\{0\right\}$ and $x,y\in X$. After improving and supplementing some former results on linear relations, we show that a relation $\Phi$ of a linearly independent subset $E$ of $X$ into $Y$ can be extended to a linear relation $F$ of $X$ into $Y$ if and only if there exists a linear subspace $Z$ of $Y$ such that $\Phi \left(e\right)\in Y|Z$ for all $e\in E$. Moreover, if...

Let $\left(\genfrac{}{}{0pt}{}{I}{m}\right)$ be the set of subsets of $I$ of cardinality $m$. Let $f$ be a coloring of $\left(\genfrac{}{}{0pt}{}{I}{m}\right)$ and $g$ a coloring of $\left(\genfrac{}{}{0pt}{}{I}{m}\right)$. We write $f\to g$ if every $f$-homogeneous $H\subseteq I$ is also $g$-homogeneous. The least $m$ such that $f\to g$ for some $f:\left(\genfrac{}{}{0pt}{}{I}{m}\right)\to k$ is called the of $g$ and denoted by $w_k\left(g\right)$. In the first part of the paper we prove the existence of colorings with high $k$-width. In particular, we show that for each $k>0$ and $m>0$ there is a coloring $g$ with $w_k\left(g\right)=m$. In the second part of the paper we give applications of wide colorings in the theory of generalized...