Threedimensional $C_{12}$manifolds.
Gherici Beldjilali
The present paper is devoted to threedimensional $C_{12}$manifolds (defined by D. Chinea
and C. Gonzalez), which are never normal. We study their fundamental properties and give
concrete examples. As an application, we study such structures on threedimensional Lie
groups.

1–14 
On an extension of the Newton polygon test for polynomial reducibility.
Brahim Boudine
Let $R$ be a commutative local principal ideal ring which is not integral, $f$ a
polynomial in $R[x]$ such that $f(0) \neq 0$ and $N(f)$ its Newton polygon. If $N(f)$
contains $r$ sides of different slopes, we show that $f$ has at least $r$ different pure
factors in $R[x]$. This generalizes the Newton polygon method over a ring which is not
integral.

15–25 
Summing the largest prime factor over integer sequences.
JeanMarie De Koninck and Rafael Jakimczuk
Given an integer $n\ge 2$, let $P(n)$ stand for its largest prime factor. We examine the
behaviour of $\sum\limits_{n\le x \atop n\in A} P(n)$ in the case of two sets $A$, namely
the set of $r$free numbers and the set of $h$full numbers.

27–35 
New classes of statistical manifolds with a complex structure.
Mirjana Milijević
We define new classes of statistical manifolds with a complex structure. Motivation for
our work is the classification of almost Hermitian manifolds with respect to the covariant
derivative of the almost complex structure, obtained by Gray and Hervella in 1980. Instead
of the LeviCivita connection, we use a statistical one and obtain eight classes of Kähler
manifolds with the statistical connection. Besides, we give some properties of tensors
constructed from covariant derivative of the almost complex structure with respect to the
statistical connection. From the obtained properties, further investigation of statistical
manifolds is possible.

37–45 
Coordinate rings of some $\mathrm{SL}_2$character varieties.
Vicente Muñoz and Jesús Martín Ovejero
We determine generators of the coordinate ring of $\mathrm{SL}_2$character varieties. In
the case of the free group $F_3$ we obtain an explicit equation of the
$\mathrm{SL}_2$character variety. For free groups $F_k$, we find transcendental
generators. Finally, for the case of the $2$torus, we get an explicit equation of the
$\mathrm{SL}_2$character variety and use the description to compute their
$E$polynomials.

47–64 
Spectrality of planar Moran–Sierpinskitype measures.
Qian Li and MinMin Zhang
Let $\{M_n\}_{n=1}^{\infty}$ be a sequence of expanding positive integral matrices with
$M_n= \begin{pmatrix} p_n & 0\\0 & q_n \end{pmatrix}$ for each $n\ge 1$, and let
$D=\left\{\begin{pmatrix} 0\ 0 \end{pmatrix}, \begin{pmatrix} 1\ 0 \end{pmatrix},
\begin{pmatrix} 0\ 1 \end{pmatrix} \right\}$ be a finite digit set in $\mathbb{Z}^2$. The
associated Borel probability measure obtained by an infinite convolution of atomic
measures \[
\mu_{\{M_n\},D}=\delta_{M_1^{1}D}*\delta_{(M_2M_1)^{1}D}*\cdots*\delta_{(M_n\cdots
M_2M_1)^{1}D}*\cdots \] is called a Moran–Sierpinskitype measure. We prove that, under
certain conditions, $\mu_{\{M_n\}, D}$ is a spectral measure if and only if $3\mid p_n$
and $3\mid q_n$ for each $n\geq2$.

65–80 
Using digraphs to compute determinant, permanent, and Drazin inverse of circulant matrices with two parameters.
Andrés M. Encinas, Daniel A. Jaume, Cristian Panelo, and Denis E. Videla
This work presents closed formulas for the determinant, permanent, inverse, and Drazin
inverse of circulant matrices with two nonzero coefficients.

81–106 
Existence and multiplicity of solutions for $p$Kirchhofftype Neumann problems.
Qin Jiang, Sheng Ma, and Daniel Paşca
We establish, based on variational methods, existence theorems for a $p$Kirchhofftype
Neumann problem under the Landesman–Lazer type condition and under the local coercive
condition. In addition, multiple solutions for a $p$Kirchhofftype Neumann problem are
established using a known threecriticalpoint theorem proposed by H. Brezis and L.
Nirenberg.

107–121 
Poincaré duality for Hopf algebroids.
Sophie Chemla
We prove a twisted Poincaré duality for (full) Hopf algebroids with bijective antipode. As
an application, we recover the Hochschild twisted Poincaré duality of van den Bergh. We
also get a Poisson twisted Poincaré duality, which was already stated for oriented Poisson
manifolds by Chen et al.

123–136 
Gorenstein properties of splitbynilpotent extension algebras.
Pamela Suarez
Let $A$ be a finitedimensional $k$algebra over an algebraically closed field $k$. In
this note, we study the Gorenstein homological properties of a splitbynilpotent
extension algebra. Let $R$ be a splitbynilpotent extension of $A$. We provide sufficient
conditions to ensure when a Gorensteinprojective module over $A$ induces a similar
structure over $R$. We also study when a Gorensteinprojective $R$module induces a
Gorensteinprojective $A$module. Moreover, we study the relationship between the
Gorensteinness of $A$ and $R$.

137–144 