We give an operator space characterization of subalgebras of $C(\Omega,M_n)$. We also describe injective subspaces of $C(\Omega,M_n)$ and then give applications to sub-TROs of $C(\Omega,M_n)$. Finally, we prove an '$n$-minimal version' of the Christensen-Effros-Sinclair representation theorem.

This paper concerns convex functions that arise as potentials of quasiconformal mappings. Several equivalent definitions for such functions are given. We use them to construct quasiconformal mappings whose Jacobian determinants are singular on a prescribed set of Hausdorff dimension less than 1.

Let $K$ be a number field. We give an arithmetic characterization at infinity of the differential operator of $K[x,d/dx]$ with minimal degree in $x$ annihilating a given $E$-function. Such an operator is called an $E$-operator.

We show that for every non-negative integer $d$, there exist differential equations $w''+Pw=0$, where $P$ is a polynomial of degree $d$, such that some non-trivial solution $w$ has all roots real.

We prove $BMO$ and $L^\infty$ results for operators associated to the heat-diffusion and Poisson semigroups in the multi-dimensional Hermite function expansions setting. These include maximal functions and square function operators. In the proof a technique of vector valued Calderón-Zygmund operators is used.

We study domain monotonicity of the principal eigenvalue $\lambda_1^\Omega(\alpha)$ corresponding to $\Delta u=\lambda(\alpha) \, u \text{ in } \Omega, \frac{\partial u}{\partial \nu} =\alpha\, u \text{ on } \partial \Omega$, with $\Omega \subset {\mathcal R}^n$ a $C^{0,1}$ bounded domain, and $\alpha$ a fixed real. We show that contrary to intuition domain monotonicity might hold if one of the two domains is a ball.

We study the boundary behavior of bounded quasiregular mappings $f: B^n(0,1)\to \rn$, $n \geq 3$. We show that there exists a large family of cusps, with vertices on the boundary sphere $S^{n-1}(0,1)$, so that the images of these cusps under $f$ have finite $(n-1)$-measure.