We prove the boundedness of several classes of rough integral operators on Triebel-Lizorkin spaces. Our results represent improvements as well as natural extensions of many previously known results.

Quadruples $(a,b,c,d)$ of positive integers $a<b<c<d$ with the property that the product of any two of them is one more than a perfect square are studied. Improved lower and upper bounds for the entries $b$ and $c$ are established. As an application of these results, a bound for the number of such quadruples is obtained.

We consider a planar surface $\Sigma$ of infinite type which has Thompson's group $\mathcal{T}$ as asymptotic mapping class group. We construct the asymptotic pants complex $\mathcal{C}$ of $\Sigma$ and prove that the group $\mathcal{T}$ acts transitively by automorphisms on it. Finally, we establish that the automorphism group of the complex $\mathcal{C}$ is an extension of the Thompson group $\mathcal{T}$ by $\mathbb{Z}/2\mathbb{Z}$

We consider obstacle problems with measure data related to elliptic equations of $p$-Laplace type, and investigate the connections between low order regularity properties of the solutions and non-linear potentials of the data. In particular, we give pointwise estimates for the solutions in terms of Wolff potentials and address the questions of boundedness and continuity of the solution.

We establish a fixed point theorem for group actions on $L^p$-spaces, which generalizes a theorem of Żuk and of Ballmann-Świątkowski to the case $p \neq 2$.

For certain contractible $G$-CW-complexes and $\mathfrak F$ a family of subgroups of $G$, we construct a spectral sequence converging to the $\mathfrak F$-Bredon cohomology of $G$ with $\mathrm{E}_1$-terms given by the $\mathfrak F$-Bredon cohomology of the stabilizer subgroups. As applications, we obtain several corollaries concerning the cohomological and geometric dimensions of the classifying space $E_{\mathfrak {F}}G$. We also introduce, for any subgroup closed class of groups $\mathfrak F$, a hierarchically defined class of groups and show that if a group $G$ is in this class, then $G$ has finite $\mathfrak F\cap G$-Bredon (co)homological dimension if and only if $G$ has jump $\mathfrak F\cap G$-Bredon (co)homology.

For all but one positive integer triplet $(a,b,c)$ with $a\leqslant b\leqslant c$ and $b\leqslant 6$, we decide whether there are algebraic numbers $\alpha$, $\beta$ and $\gamma$ of degrees $a$, $b$ and $c$, respectively, such that $\alpha+\beta+\gamma=0$. The undecided case $(6,6,8)$ will be included in another paper. These results imply, for example, that the sum of two algebraic numbers of degree $6$ can be of degree $15$ but cannot be of degree $10$. We also show that if a positive integer triplet $(a,b,c)$ satisfies a certain triangle-like inequality with respect to every prime number then there exist algebraic numbers $\alpha$, $\beta$, $\gamma$ of degrees $a$, $b$, $c$ such that $\alpha+\beta+\gamma=0$. We also solve a similar problem for all $(a,b,c)$ with $a\leqslant b\leqslant c$ and $b\leqslant 6$ by finding for which $a$, $b$, $c$ there exist number fields of degrees $a$ and $b$ such that their compositum has degree $c$. Further, we have some results on the multiplicative version of the first problem, asking for which triplets $(a,b,c)$ there are algebraic numbers $\alpha$, $\beta$ and $\gamma$ of degrees $a$, $b$ and $c$, respectively, such that $\alpha\beta\gamma=1$.

The pseudovariety $\mathbf{PCS}$ which is generated by all power semigroups of finite completely simple semigroups is characterized in various ways. For example, the equalities

$$\mathbf{PCS}=\mathbf{J}\malcab \mathbf{CS} =\mathbf{BG}\malcab \mathbf{RB}$$

are established. This resolves a problem raised by Kaďourek and leads to several transparent algorithms for deciding membership in $\mathbf{PCS}$.

Lower bounds for the number of elements of the largest non-commuting set of a finite soluble group with a CC-subgroup are considered in this paper