1.
A characterization of hyperbolic Kato surfaces - Brunella, Marco
We give a characterization of hyperbolic Kato surfaces in terms of the existence of an automorphic Green function on a cyclic covering. This is achieved by analysing a naturally defined Levi-flat foliation,
and by perturbing certain Levi-flat leaves to strictly pseudoconvex hypersurfaces.

2.
On rings with finite number of orbits - Hryniewicka, Małgorzata; Krempa, Jan
Let $R$ be an associative unital ring with the unit group $U(R)$. Let $\mathcal{S}$ denote one of the following sets: the set of elements of $R$, of left ideals of $R$, of principal left
ideals of $R$, or of ideals of $R$. Then the group $U(R)\times U(R)$ acts on the set $\mathcal{S}$ by left and right multiplication. In this note we are going to discuss some properties
of rings $R$ with a finite number of orbits under the action of $U(R)\times U(R)$ on $\mathcal{S}$.

4.
Non-existence of multi-line Besicovitch sets - Orponen, Tuomas
If a compact set $K \subset \mathbb{R}^{2}$ contains a positive-dimensional family of line-segments in every direction, then $K$ has positive measure.

5.
Generalized quasidisks and conformality - Guo, Chang-Yu; Koskela, Pekka; Takkinen, Juhani
We introduce a weaker variant of the concept of linear local connectivity, sufficient to guarantee the extendability of a conformal map $f\colon\mathbb D\to\Omega$ to the entire
plane as a homeomorphism of locally exponentially integrable distortion. Additionally, we show that a conformal map as above cannot necessarily be extended in this manner if we assume
that $\Omega$ is the image of $\mathbb D$ under a self-homeomorphism of the plane that has locally exponentially integrable distortion.

6.
Submanifolds with nonparallel first normal bundle revisited - Dajczer, Marcos; Tojeiro, Ruy
In this paper, we analyze the geometric structure of a Euclidean submanifold whose
osculating spaces form a nonconstant family of proper subspaces
of the same dimension. We prove that if the rate of change of the osculating spaces
is small, then the submanifold must be a (submanifold of a) ruled submanifold of a
very special type. We also give a sharp estimate of the dimension of the rulings.

7.
Dilations and full corners on fractional skew monoid rings - Pardo, E.
In this note we will show that the dilation result obtained for fractional skew monoid rings, in the case of a cancellative left Ore
monoid $S$ acting on a unital ring $A$ by corner isomorphisms, holds in full generality. We apply this result to the context of semigroup $C^*$-crossed products.

8.
Weak and viscosity solutions of the fractional Laplace equation - Servadei, Raffaella; Valdinoci, Enrico
Aim of this paper is to show that weak solutions of the following fractional Laplacian equation
$$(-\Delta)^s u=f &\Omega\\u=g &\mathbb R^n\setminus\Omega$$
¶ are also continuous solutions (up to the boundary) of this problem in the viscosity sense.
¶ Here $s\in(0,1)$ is a fixed parameter, $\Omega$ is a bounded, open subset of $\mathbb R^n$ ($n\geqslant1$)
with $C^2$-boundary, and $(-\Delta)^s$ is the fractional Laplacian operator, that may be defined as
$$(-\Delta)^su(x):=c(n,s)\int\limits_{\mathbb R^n}\frac{2u(x)-u(x+y)-u(x-y)}{|y|^{n+2s}}\,dy,$$
¶ for a suitable positive normalizing constant $c(n,s)$, depending only on $n$ and $s$.
¶ In order to get our regularity result we first prove a maximum principle and then, using it, an interior and boundary regularity...

10.
Dynamics of (pseudo) automorphisms of 3-space: periodicity versus positive entropy - Bedford, Eric; Kim, Kyounghee
We study the iteration of the family of maps given by $3$-step linear fractional recurrences. This family was studied earlier from the point of view of finding periodicities. In this paper we finish that study by
determining all possible periods within this family. The novelty of our approach is that we apply the methods of complex dynamical systems. This leads to two classes of interesting pseudo automorphisms
of infinite order. One of the classes consists of completely integrable maps. The other class consists of maps of positive entropy which have an invariant family of $K3$ surfaces.

11.
Revisiting the Fourier transform on the Heisenberg group - Lavanya, R. Lakshmi; Thangavelu, S.
A recent theorem of S. Alesker, S. Artstein-Avidan and V. Milman characterises the Fourier transform on $ {\mathbb R}^{n} $ as essentially the only transform on the space of tempered distributions
which interchanges convolutions and pointwise products. In this note we study the image of the Schwartz space on the Heisenberg
group under the Fourier transform and obtain a similar characterisation for the Fourier transform on the Heisenberg group

12.
The oriented graph of multi-graftings in the Fuchsian case - Calsamiglia, Gabriel; Deroin, Bertrand; Francaviglia, Stefano
We prove the connectedness and compute the diameter of the oriented graph of multi-graftings associated to exotic
$\mathbb{CP}^1$-structures on a compact surface~$S$ with a given holonomy representation of Fuchsian type.

13.
Marcinkiewicz interpolation theorems for Orlicz and Lorentz gamma spaces - Kerman, Ron; Phipps, Colin; Pick, Lubosš
Fix the indices $\alpha$ and $\beta$, $1<\alpha<\beta<\infty$, and suppose $\varrho$ is an Orlicz gauge or Lorentz gamma norm on the real-valued functions on a set $X$ which
are measurable with respect to a~$\sigma$-finite measure $\mu$ on it. Set
$$M(\gamma,X):=\{f\colon X\to\mathbb R \text{ with } \sup_{\lambda>0}\lambda \mu(\{x\in X: |f(x)|>\lambda\})^{\frac1{\gamma}}<\infty\},$$
¶ $\gamma=\alpha,\beta$. In this paper we obtain, as a special case, simple criteria to guarantee that a linear operator $T$ satisfies
$T\colon L_{\varrho}(X)\to L_{\varrho}(X)$, whenever $T\colon M(\alpha,X)\to M(\alpha, X)$ and $T\colon M(\beta,X)\to M(\beta, X)$.

14.
Vanishing results for the cohomology of complex toric hyperplane complements - Davis, M. W.; Settepanella, S.
Suppose $\mathcal R$ is the complement of an essential arrangement of toric
hyperlanes in the complex torus $(\mathbb{C}^*)^n$ and $\pi=\pi_1(\mathcal
R)$. We show that $H^*(\mathcal R;A)$ vanishes except in the top degree $n$
when $A$ is one of the following systems of local coefficients: (a) a system
of nonresonant coefficients in a complex line bundle, (b) the von Neumann
algebra $\mathcal{N}\pi$, or (c) the group ring ${\mathbb Z} \pi$. In case
(a) the dimension of $H^n$ is $|e(\mathcal R)|$ where $e(\mathcal R)$
denotes the Euler characteristic, and in case (b) the $n^{\mathrm{th}}$
$\ell^2$ Betti number is also $|e(\mathcal R)|$

15.
Entropy and Flatness in Local Algebraic Dynamic - Majidi-Zolbanin, Mahdi; Miasnikov, Nikita; Szpiro, Lucien
For a local endomorphism of a noetherian local ring we introduce a notion of
entropy, along with two other asymptotic invariants. We use this notion of
entropy to extend numerical conditions in Kunz' regularity criterion to every
contracting endomorphism of a noetherian local ring, and to give a
characteristic-free interpretation of the definition of Hilbert-Kunz
multiplicity. We also show that every finite endomorphism of a complete
noetherian local ring of equal characteristic can be lifted to a finite
endomorphism of a complete regular local ring. The local ring of an algebraic or
analytic variety at a point fixed by a finite self-morphism inherits a local
endomorphism whose entropy is...

16.
An Extension of Sub-Fractional Brownian Motion - Sghir, Aissa
In this paper, firstly, we introduce and study a self-similar Gaussian process
with parameters $H \in{(0,1)}$ and $K \in(0,1]$ that is an extension of the well
known sub-fractional Brownian motion introduced by Bojdecki et al.
Secondly, by using a decomposition in law of this process, we prove the
existence and the joint continuity of its local time

17.
Conjugacy classes of left ideals of a finite dimensional
algebra - Mȩcel, Arkadiusz; Okniński, Jan
Let $A$ be a finite dimensional unital algebra over a field $K$ and let $C(A)$
denote the set of conjugacy classes of left ideals in $A$. It is shown that
$C(A)$ is finite if and only if the number of conjugacy classes of nilpotent
left ideals in $A$ is finite. The set~$C(A)$ can be considered as a semigroup
under the natural operation induced from the multiplication in $A$. If $K$ is
algebraically closed, the square of the radical of~$A$ is zero and $C(A)$ is
finite, then for every $K$-algebra $B$ such that $C(B)\cong C(A)$ it is shown
that $B\cong A$.

18.
Blei's Inequality and Coordinatewise Multiple Summing
Operators - Popa, Dumitru; Sinnamon, Gord
Two inequalities resembling the multilinear Hölder inequality for mixed-norm
Lebesgue spaces are proved. When specialized to single-function inequalities
they include a pair of inequalities due to Blei and a recent extension of Blei's
inequality. The first of these inequalities is applied to give explicit indices
in a known result for coordinatewise multiple summing operators. The second is
used to prove a complementary result to the known one, again with explicit
indices. As an application of the complementary result, a sufficient condition
is given for a composition of operators to be multiple summing.

19.
Layer potentials beyond singular integral operator - Rosén, Andreas
We prove that the double layer potential operator and the gradient of the single
layer potential operator are $L_2$ bounded for general second order divergence
form systems. As compared to earlier results, our proof shows that the bounds
for the layer potentials are independent of well posedness for the Dirichlet
problem and of De Giorgi-Nash local estimates. The layer potential operators are
shown to depend holomorphically on the coefficient matrix $A\in L_\infty$,
showing uniqueness of the extension of the operators beyond singular integrals.
More precisely, we use functional calculus of differential operators with
non-smooth coefficients to represent the layer potential operators as bounded
Hilbert space operators. In the presence...

20.
Polygonal $\mathcal{VH}$ Complexes - Polák, Jason K. C.; Wise, Daniel T.
Ian Leary inquires whether a class of hyperbolic finitely presented groups are
residually finite. We answer in the affirmative by giving a systematic version
of a construction in his paper, which shows that the standard $2$-complexes of
these presentations have a $\mathcal{VH}$-structure. This structure induces a
splitting of these groups, which together with hyperbolicity, implies that these
groups are residually finite.