The norm problem is considered for elementary operators of the form $U_{a,b}\,{:}\,{\cal A}\,{\longrightarrow}\,{\cal A},\;x\longmapsto axb\,{+}\,bxa (a,\,b\,{\in}\,{\cal A})$ in the special case when ${\cal A}$ is a subalgebra of the algebra of bounded operators on a Banach space. In particular, the lower estimate $\|U_{a,b}\|\geq\|a\|\|b\|$ is established when the Banach space is a Hilbert space and ${\cal A}$ is the algebra of all bounded linear operators.

It is established that in a commutative radical Fréchet algebra, elements of non-nilpotent finite closed descent exist if a locally non-nilpotent element of locally finite closed descent exists. Thus if $\mathbb{C}[[X]]$ can be embedded into the unitization of the algebra in such a way that $X$ is mapped to an element which is locally non-nilpotent, then it is possible to embed the ‘structurally rich’ algebra $\mathbb{C}_{\omega_{1}}$.

Hoh's calculus for pseudo-differential operators with negative-definite symbols is extended to the case of parameter-dependent symbols. Then this parameter-dependent calculus is applied to the study of subordinate sub-Markovian semigroups.

Bounded, compact and Schatten class weighted composition operators on the Bergman space are characterized by the use of generalized Berezin transforms. An estimate of the essential norms of weighted composition operators on the Bergman space is given. Most of the results remain true for Hardy spaces and weighted Bergman spaces.

The paper considers new components of a key space of a moduli space of minimal surfaces in flat 4-tori and calculates their dimensions. An example of minimal surfaces in 4-tori is constructed, and an element of the moduli is obtained. In the process of construction, an example is given of minimal surfaces with a good property in a 3-torus that is distinct from classical examples.

It is shown that every strongly-cyclic branched covering of a (1,1)-knot is a Dunwoody manifold. This result, together with the converse statement previously obtained by Grasselli and Mulazzani, proves that the class of Dunwoody manifolds coincides with the class of strongly-cyclic branched coverings of (1,1)-knots. As a consequence, a parametrization of (1,1)-knots by 4-tuples of integers is obtained. Moreover, using a representation of (1,1)-knots by the mapping class group of the twice-punctured torus, an algorithm is provided which gives the parametrization of all torus knots in $\mathbf{S}^3$.

For a convex body $K$ in $\R^d$ and $1\le k\le d-1$, let $P_k(K)$ be the Minkowski sum (average) of all orthogonal projections of $K$ onto $k$-dimensional subspaces of $\R^d$. It is known that the operator $P_k$ is injective if $k\,{\geq}\, d/2$, $k\,{=}\,3$ for all $d$, and if $k=2$, $d\ne 14$.

It is shown that $P_{2k}(K)$ determines a convex body $K$ among all centrally symmetric convex bodies and $P_{2k+1}(K)$ determines a convex body $K$ among all bodies of constant width. Corresponding stability results are also given. Furthermore, it is shown that any convex body $K$ is determined by the two sets $P_k(K)$ and $P_{k'}(K)$ if $1<k<k'$. Concerning the range of $P_k$, $1\le k\le d-2$, it is shown that its closure (in the Hausdorff-metric) does not contain any polytopes other than singletons.