Given a closed Riemann surface $R$ of genus $p \geq 2$ together with an anticonformal involution $\tau:R \to R$ with fixed points, we consider the group $K(R,\tau)$ consisting of the conformal and anticonformal automorphisms of $R$ which commute with $\tau$. It is a well known fact due to C. L. May that the order of $K(R,\tau)$ is at most $24(p-1)$ and that such an upper bound is attained for infinitely many, but not all, values of $p$. May also proved that for every genus $p \geq 2$ there are surfaces for which the order of $K(R,\tau)$ can be chosen to be $8p$ and $8(p+1)$. These type of surfaces are called \textit{May surfaces}. In this note we construct real Schottky uniformizations of every May surface. In particular, the corresponding group $K(R,\tau)$ lifts to such an uniformization. With the help of these real Schottky uniformizations, we obtain (extended) symplectic representations of the groups $K(R,\tau)$. We study the families of principally polarized abelian varieties admitting the given group of automorphisms and compute the corresponding Riemann matrices, including those for the Jacobians of May surfaces.

This paper is devoted to Banach algebras generated by Toeplitz operators with strongly oscillating symbols, that is, with symbols of the form $b(e^{i\alpha(x)})$ where $b$ belongs to some algebra of functions on the unit circle and $\alpha$ is a fixed orientation-preserving homeomorphism of the real line onto itself. We prove the existence of certain interesting homomorphisms and establish conditions for the normal solvability, Fredholmness, and invertibility of operators in these algebras.

We study regularity properties of a positive measure in the euclidean space in terms of two square functions which are the multiplicative analogues of the usual martingale square function and of the Lusin area function of a harmonic function. The size of these square functions is related to the rate at which the measure doubles at small scales and determines several regularity properties of the measure. We consider the non-tangential maximal function of the logarithm of the densities of the measure in the dyadic setting, and of the logarithm of the harmonic extension of the measure, in the continuous setting. We relate the size of these maximal functions to the size of the corresponding square functions. Fatou type results, $L^p$ estimates and versions of the Law of the Iterated Logarithm are proved. As applications we introduce a hyperbolic version of the Lusin Area function of an analytic mapping from the unit disc into itself, and use it to characterize inner functions. Another application to the theory of quasiconformal mappings is given showing that our methods can also be applied to prove a result by Din'kyn's on the smoothness of quasiconformal mappings of the disc.

Let $S$ be a real closed Riemann surfaces together a reflection \mbox{$\tau:S \to S$}, that is, an anticonformal involution with fixed points. A well known fact due to C. L. May \cite{May 1977} asserts that the group $K(S,\tau)$, consisting on all automorphisms (conformal and anticonformal) of $S$ which commutes with $\tau$, has order at most $24(g-1)$. The surface $S$ is called maximally symmetric Riemann surface if $|K(S,\tau)|=24(g-1)$ \cite{Greenleaf-May 1982}. In this note we proceed to construct real Schottky uniformizations of all maximally symmetric Riemann surfaces of genus $g \leq 5$. A method due to Burnside \cite{Burnside 1892} permits us the computation of a basis of holomorphic one forms in terms of these real Schottky groups and, in particular, to compute a Riemann period matrix for them. We also use this in genus 2 and 3 to compute an algebraic curve representing the uniformized surface $S$. The arguments used in this note can be programed into a computer program in order to obtain numerical approximation of Riemann period matrices and algebraic curves for the uniformized surface $S$ in terms of the parameters defining the real Schottky groups.

The analysis of dissipative transport equations within the framework of open quantum systems with Fokker-Planck-type scattering is carried out from the perspective of a Wigner function approach. In particular, the well-posedness of the self-consistent whole-space problem in 3D is analyzed: existence of solutions, uniqueness and asymptotic behavior in time, where we adopt the viewpoint of mild solutions in this paper. Also, the admissibility of a density matrix formulation in Lindblad form with Fokker-Planck dissipation mechanisms is discussed. We remark that our solution concept allows to carry out the analysis directly on the level of the kinetic equation instead of on the level of the density operator.

We study spherical pulse like families of solutions to semilinear wave equations in space time of dimension 1+3 as the pulses focus at a point and emerge outgoing. We emphasize the scales for which the incoming and outgoing waves behave linearly but the nonlinearity has a strong effect at the focus. The focus crossing is described by a scattering operator for the semilinear equation, which broadens the pulses. The relative errors in our approximate solutions are small in the $L^\infty$ norm.

Consider a system consisting of a linear wave equation coupled to a transport equation: \begin{equation*} \Box_{t,x}u =f , \end{equation*} \begin{equation*} (\partial_t + v(\xi) \cdot \nabla_x)f =P(t,x,\xi, D_\xi)g , \end{equation*} Such a system is called \textit{nonresonant} when the maximum speed for particles governed by the transport equation is less than the propagation speed in the wave equation. Velocity averages of solutions to such nonresonant coupled systems are shown to be more regular than those of either the wave or the transport equation alone. This smoothing mechanism is reminiscent of the proof of existence and uniqueness of $C^1$ solutions of the Vlasov-Maxwell system by R. Glassey and W. Strauss for time intervals on which particle momenta remain uniformly bounded, see ``Singularity formation in a collisionless plasma could occur only at high velocities'', \textit{Arch. Rational Mech. Anal.} \textbf{92} (1986), no. 1, 59-90. Applications of our smoothing results to solutions of the Vlasov-Maxwell system are discussed.

Given a real-valued random variable $X$ whose Laplace transform is analytic in a neighbourhood of 0, we consider a random walk ${(S_{n},n\geq 0)}$, starting from the origin and with increments distributed as $X$. We investigate the class of stopping times $T$ which are independent of $S_{T}$ and standard, i.e. $(S_{n\wedge T},n\geq 0)$ is uniformly integrable. The underlying filtration $(\mathcal{F}_{n},n\geq 0)$ is not supposed to be natural. Our research has been deeply inspired by \cite{De Meyer-Roynette-Vallois-Yor 2002}, where the analogous problem is studied, but not yet solved, for the Brownian motion. Likewise, the classification of all possible distributions for $S_{T}$ remains an open problem in the discrete setting, even though we manage to identify the solutions in the special case where $T$ is a stopping time in the natural filtration of a Bernoulli random walk and $\min T \le 5$. Some examples illustrate our general theorems, in particular the first time where $|S_{n}|$ (resp. the age of the walk or Pitman's process) reaches a given level $a\in\mathbb{N}^{\ast}$. Finally, we are concerned with a related problem in two dimensions. Namely, given two independent random walks $(S_{n}^{\prime},n\geq 0)$ and $(S_{n}^{\prime\prime},n\geq 0)$ with the same incremental distribution, we search for stopping times $T$ such that $S_{T}^{\prime}$ and $S_{T}^{\prime\prime}$ are independent.

We give an explicit description of a non-normal irreducible subvariety of the moduli space of Riemann surfaces of genus $3$ characterized by a non-cyclic group action. Defining equations of a family of curves representing non-normal points of this subvariety are computed. We also find defining equations of the family of hyperelliptic curves of genus $3$ whose full automorphism group is $C_2\times C_4$. This completes the list of full automorphism groups of hyperelliptic curves.