1.
Some Local Properties Defining ${\mathcal T}_0$-Groups and Related Classes of Groups - Ballester-Bolinches, A.; Beidleman, J.C.; Esteban-Romero, R.; Ragland, M.F.
We call $G$ a $\operatorname{Hall}_{\mathcal X}$-group if there exists a normal nilpotent subgroup $N$
of $G$ for which $G/N'$ is an ${\mathcal X}$-group. We call $G$ a ${\mathcal T}_0$-group
provided $G/\Phi(G)$ is a ${\mathcal T}$-group, that is, one in which normality is a transitive relation. We present several new local classes of groups which locally define
$\operatorname{Hall}_{\mathcal X}$-groups and
${\mathcal T}_0$-groups where ${\mathcal X}\in\{ {\mathcal T},\mathcal {PT},\mathcal {PST}\}$;
the classes $\mathcal {PT}$ and $\mathcal {PST}$ denote, respectively, the classes of groups in which permutability and S-permutability are transitive relations.

2.
Extreme Cycles. The Center of a Leavitt Path Algebra - Corrales García, María G.; Martín Barquero, Dolores; Martín González, Cándido; Siles Monlina, Mercedes; Solanilla Hernández, José F.
In this paper we introduce new techniques in order to deepen into the structure of a Leavitt path algebra with the aim of giving a description of the center. Extreme cycles appear for the first time; they
concentrate the purely infinite part of a Leavitt path algebra and, jointly with the line points and vertices in cycles without exits, are the key ingredients in order to determine the center of a Leavitt path
algebra. Our work will rely on our previous approach to the center of a prime Leavitt path algebra, "Centers of path algebras, Cohn and Leavitt path algebras,"
Bull. Malays. Math....

3.
On the Galois Correspondence Theorem in Separable Hopf Galois Theory - Crespo, Teresa; Rio, Anna; Vela, Montserrat
In this paper we present a reformulation of the Galois
correspondence theorem of Hopf Galois theory in terms of groups
carrying farther the description of Greither and Pareigis. We
prove that the class of Hopf Galois extensions for which the
Galois correspondence is bijective is larger than the class of
almost classically Galois extensions but not equal to the whole
class. We show as well that the image of the Galois correspondence
does not determine the Hopf Galois structure.

4.
Upper Bound for Multi-Parameter Iterated Commutators - Dalenc, Laurent; Ou, Yumeng
We show that the product BMO space can be characterized by iterated commutators of a large class of Calderón--Zygmund operators. This result follows from a new proof of boundedness of iterated
commutators in terms of the BMO norm of their symbol functions, using Hytönen's representation theorem of Calderón--Zygmund operators as averages of dyadic shifts. The proof introduces
some new paraproducts which have BMO estimates.

5.
Mixed norm estimates for the Riesz transforms on \boldmath$SU(2)$ - Boggarapu, Pradeep; Thangavelu, S.
In this paper we prove mixed norm estimates for Riesz
transforms on the group $SU(2)$. From these results vector valued inequalities for sequences of Riesz transforms associated to Jacobi differential operators of different types are deduced.

6.
Fine Gradings on $\mathfrak e_6$ - Draper, Cristina; Viruel, Antonio
There are fourteen fine gradings on the exceptional Lie algebra $\mathfrak e_6$ over
an algebraically closed field of zero characteristic. We provide their descriptions and
a proof that any fine grading is equivalent to one of them.

7.
Geometric Characterizations of $p$-Poincaré Inequalities in the Metric Setting - Durand-Cartagena, Estibalitz; Jaramillo, Jesus A.; Shanmugalingam, Nageswari
We prove that a locally complete metric space endowed with a doubling measure satisfies an $\infty$-Poincaré inequality if and only if given a null set, every two points
can be joined by a quasiconvex curve
which "almost avoids" that set. As an application, we characterize doubling measures on ${\mathbb R}$
satisfying an $\infty$-Poincaré inequality. For Ahlfors $Q$-regular spaces,
we obtain a characterization of $p$-Poincaré inequality for $p>Q$ in terms of the $p$-modulus of quasiconvex
curves connecting pairs of points in the space. A related characterization is given for the case $Q-1

8.
Nilpotent Groups of Class Three and Braces - Cedó, Ferran; Jespers, Eric; OkniŃski, Jan
New constructions of braces on finite nilpotent groups are given and hence this leads to new solutions of the Yang--Baxter equation. In
particular, it follows that if a group $G$ of odd order is
nilpotent of class three, then it is a homomorphic image of the
multiplicative group of a finite left brace (i.e.\ an involutive
Yang--Baxter group) which also is a nilpotent group of class three. We
give necessary and sufficient conditions for an arbitrary group $H$
to be the multiplicative group of a left brace such that $[H,H]
\subseteq \operatorname{Soc} (H)$ and $H/[H,H]$ is a standard abelian brace, where $\operatorname{Soc} (H)$ denotes the socle of...

9.
A Nonlocal 1-Laplacian Problem and Median Values - Mazón, José M.; Pérez-Llanos, Mayte; Rossi, Julio D.; Toledo, Julián
In this paper, we study solutions to a nonlocal $1$-Laplacian
equation given by
¶ $$
-\int_{\Omega_J}
J(x-y)\frac{u_\psi(y)-u(x)}{|u_\psi(y)-u(x)|}\,dy=0\quad\text{for
$x\in\Omega$},
$$
¶ with $u(x)=\psi(x)$ for $x\in
\Omega_J\setminus\overline\Omega$. We introduce two notions of solution and prove that the weaker of
the two concepts is equivalent to a nonlocal median value property, where the median is determined by a measure related to $J$. We also show that solutions in the stronger sense are
nonlocal analogues of local least gradient functions, in the sense that they minimize a nonlocal functional. In addition, we prove that solutions in the stronger sense converge to least gradient
solutions when the kernel $J$ is appropriately rescaled.

10.
Nonlocal Equations in Bounded Domains: A Survey - Ros-Oton, Xavier
In this paper we survey some results on the Dirichlet problem
\[\begin{cases}
L u =f&\text{in }\Omega \\
u=g &\text{in }\mathbb{R}^n\backslash\Omega
\end{cases}
\]
¶ for nonlocal operators of the form
\[
Lu(x)=\operatorname{PV}\int_{\mathbb{R}^n}\bigl\{u(x)-u(x+y)\bigr\}K(y)\,dy.
\]
¶ We start from the very basics, proving existence of solutions, maximum principles, and constructing some useful barriers.
Then, we focus on the regularity properties of solutions, both in the interior and on the boundary of the domain.
¶ In order to include some natural operators~$L$ in the regularity theory, we do not assume any regularity on the kernels.
This leads to some interesting features that are purely nonlocal, in the sense that they have no analogue for local equations.
¶ We...

12.
A \boldmath$p$-adic construction of ATR points on \boldmath$\mathbb{Q}$-curves - Guitart, Xavier; Masdeu, Marc
In this note we consider certain elliptic curves defined over real quadratic fields isogenous to their Galois conjugate. We give a construction of algebraic points on these curves defined over almost totally real number fields. The main ingredient is the system of Heegner points arising from Shimura curve uniformizations. In addition, we provide an explicit $p$-adic analytic formula which allows for the effective, algorithmic calculation of such points.

13.
Optimal Quasi-Metrics in a Given Pointwise Equivalence Class do not Always Exist - Brigham, Dan; Mitrea, Marius
In this paper we provide an answer to a question found in
"Groupoid metrization theory," With applications to analysis on quasi-metric spaces and functional analysis,, namely when given a quasi-metric $\rho$, if one examines all quasi-metrics which are pointwise equivalent to $\rho$, does there exist one which is most like an ultrametric (or, equivalently, exhibits an optimal amount of Hölder regularity)? The answer, in general, is negative, which we demonstrate by constructing a suitable Rolewicz--Orlicz space.

14.
Second order geometry\\ of spacelike surfaces in de Sitter 5-space - Kasedou, Masaki; Nabarro, Ana Claudia; Ruas, Maria Aparecida Soares
The de Sitter space is known as a Lorentz space with positive constant curvature in the Minkowski space. A surface with a Riemannian metric is called a spacelike surface. In this work we investigate properties of the second order geometry of spacelike surfaces in de Sitter space $S_1^5$ by using the action of $GL(2,\mathbb R)\times SO(1,2)$ on the system of conics defined by the second fundamental form. The main results are the classification of the second fundamental mapping and the description of the possible configurations of the $\mathit{LMN}$-ellipse. This ellipse gives information on the lightlike binormal directions and consequently about...

15.
Compactness of higher-order Sobolev embeddings - Slavíková, Lenka
We study higher-order compact Sobolev embeddings on a domain $\Omega \subseteq \mathbb R^n$ endowed with a probability measure $\nu$ and satisfying certain isoperimetric inequality. Given $m\in \mathbb N$, we present a condition on a pair of rearrangement-invariant spaces $X(\Omega,\nu)$ and $Y(\Omega,\nu)$ which suffices to guarantee a compact embedding of the Sobolev space $V^mX(\Omega,\nu)$ into $Y(\Omega,\nu)$. The condition is given in terms of compactness of certain one-dimensional operator depending on the isoperimetric function of $(\Omega,\nu)$. We then apply this result to the characterization of higher-order compact Sobolev embeddings on concrete measure spaces, including John domains, Maz'ya classes of Euclidean domains and...

16.
Reflection groups of the quadratic form \boldmath$-px_0^2+x_1^2+\dotsb+x_n^2$ with \boldmath$p$ prime - Mark, Alice
We present the classification of reflective quadratic forms $-px_0^2+x_1^2+\dotsb+x_n^2$ for $p$ prime. We show that for $p = 5$, it is reflective for $2\leq n\leq 8$, for $p = 7\text{ and }17$ it is reflective for $n = 2\text{ and }3$, for $p=11$ it is reflective for $p=2$, $3,\text{ and } 4$, and it is not reflective for higher values of $n$. We also show that it is non-reflective for $n > 2$ when $p = 13$, $19,\text{ and }23$. This completes the classification of these forms with $p$ prime.

17.
Tout chemin générique de hérissons réalisant un retournement de la sphère dans \boldmath$\mathbb{R}^{3}$ comprend un hérisson porteur de queues d'aronde positives - Martinez-Maure, Yves
Hedgehogs are (possibly singular and self-intersecting) hypersurfaces that describe Minkowski differences of convex bodies in $\mathbb{R}^{n+1}$. They are the natural geometrical objects when one seeks to extend parts of the Brunn--Minkowski theory to a vector space which contains convex bodies. In this paper, we prove that in every generic path of hedgehogs performing the eversion of the sphere in $\mathbb{R}^{3}$, there exists a hedgehog that has positive swallowtails. This study was motivated by an open problem raised in 1985 by Langevin, Levitt, and Rosenberg.

19.
Property of rapid decay for extensions of compactly generated groups - Garncarek, Lukasz
n the article we settle down the problem of permanence of property RD under group extensions. We show that if $1\to N\to G\to Q\to 1$ is a short exact sequence of compactly generated groups such that $Q$ has property RD, and $N$ has property RD with respect to the restriction of a word-length on $G$, then $G$ has property RD.
¶ We also generalize the result of Ji and Schweitzer stating that locally compact groups with property RD are unimodular. Namely, we show that any automorphism of a locally compact group with property RD which distorts distances subexponentially, preserves the Haar...

20.
Minimal Faithful Modules over Artinian Rings - Bergman, George M.
Let $R$ be a left Artinian ring, and $M$ a faithful left $R$-module such that no proper submodule or homomorphic image of $M$ is faithful.
¶ If $R$ is local, and $\operatorname{socle}(R)$ is central in $R$, we show that $\operatorname{length}(M/J(R)M) + \operatorname{length}(\operatorname{socle}(M))\leq \operatorname{length}(\operatorname{socle}(R))+1$.
¶ If $R$ is a finite-dimensional algebra over an algebraically closed field, but not necessarily local or having central socle, we get an inequality similar to the above, with the length of $\operatorname{socle}(R)$ interpreted as its length as a bimodule, and the summand $+1$ replaced by the Euler characteristic of a graph determined by the bimodule structure of $\operatorname{socle}(R)$....