By means of the method of block decompositions for kernel functions and some delicate estimates on Fourier transforms, the $L^p(\boldsymbol{R}^m\times\boldsymbol{R}^n\times\boldsymbol{R})$-boundedness of the multiple Marcinkiewicz integral is established along a continuous surface with rough kernel for some $p>1$. The condition on the integral kernel is the best possible for the $L^2$-boundedness of the multiple Marcinkiewicz integral operator.

The main purpose of the present paper is to show that a class of dynamical zeta functions associated with the so-called two-dimensional open billiard without eclipse have meromorphic extensions to the half-plane consisting of all complex numbers whose real parts are greater than a certain negative number. As an application, we verify that the zeta function for the length spectrum of the corresponding billiard table has the same property.

We characterize the smooth toric varieties for which the Merkurjev spectral sequence, connecting equivariant and ordinary K-theory, degenerates. We find under which conditions on the support of the fan the $E^2$ terms of the spectral sequence are invariants by subdivisions of the fan. Assuming these conditions, we describe explicitly the $E^2$ terms, linking them to the reduced homology of the fan.

We obtain the topological configurations of the lines of curvature, the asymptotic and characteristic curves on a cross-cap, in the domain of a parametrisation of this surface as well as on the surface itself.

We study, for any prime number $p$, the triviality of certain primary components of the ideal class group of the $\boldsymbol{Z}_p$-extension over the rational field. Among others, we prove that if $p$ is $2$ or $3$ and $l$ is a prime number not congruent to $1$ or $-1$ modulo $2p^2$, then $l$ does not divide the class number of the cyclotomic field of $p^u$th roots of unity for any positive integer $u$.

We construct normal del Pezzo surfaces, and regular weak del Pezzo surfaces as well, with positive irregularity $q>0$. This can happen only over nonperfect fields. The surfaces in question are twisted forms of nonnormal del Pezzo surfaces, which were classified by Reid. The twisting is with respect to the flat topology and infinitesimal group scheme actions. The twisted surfaces appear as generic fibers for Fano-Mori contractions on certain threefolds with only canonical singularities.