Revista de la
Unión Matemática Argentina

Current issue

Articles are published here before the issue is completed.

Vol. 67, no. 1 (2024)

Three-dimensional $C_{12}$-manifolds. Gherici Beldjilali
The present paper is devoted to three-dimensional $C_{12}$-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 three-dimensional Lie groups.
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.
Summing the largest prime factor over integer sequences. Jean-Marie 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.
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 Levi-Civita 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.
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.
Spectrality of planar Moran–Sierpinski-type measures. Qian Li and Min-Min 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–Sierpinski-type 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 $n\geq2$.
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 non-zero coefficients.
Existence and multiplicity of solutions for $p$-Kirchhoff-type Neumann problems. Qin Jiang, Sheng Ma, and Daniel Paşca
We establish, based on variational methods, existence theorems for a $p$-Kirchhoff-type Neumann problem under the Landesman–Lazer type condition and under the local coercive condition. In addition, multiple solutions for a $p$-Kirchhoff-type Neumann problem are established using a known three-critical-point theorem proposed by H. Brezis and L. Nirenberg.
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.
Gorenstein properties of split-by-nilpotent extension algebras. Pamela Suarez
Let $A$ be a finite-dimensional $k$-algebra over an algebraically closed field $k$. In this note, we study the Gorenstein homological properties of a split-by-nilpotent extension algebra. Let $R$ be a split-by-nilpotent extension of $A$. We provide sufficient conditions to ensure when a Gorenstein-projective module over $A$ induces a similar structure over $R$. We also study when a Gorenstein-projective $R$-module induces a Gorenstein-projective $A$-module. Moreover, we study the relationship between the Gorensteinness of $A$ and $R$.