Revista de la Unión Matemática Argentina

Three-dimensional $C_{12}$-manifolds

Gherici Beldjilali — Wed, 14 Feb 2024 00:00:00 -0300

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 — Wed, 21 Feb 2024 00:00:00 -0300

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, Rafael Jakimczuk — Wed, 21 Feb 2024 00:00:00 -0300

Given an integer $n\ge 2$, let $P(n)$ stand for its largest prime factor. We examine the behaviour of $\displaystyle{\sum_{n\le x \atop n\in A} P(n)}$ 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 — Wed, 06 Mar 2024 00:00:00 -0300

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, Jesús Martín Ovejero — Fri, 08 Mar 2024 00:00:00 -0300

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, Min-Min Zhang — Tue, 12 Mar 2024 00:00:00 -0300

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, Denis E. Videla — Tue, 12 Mar 2024 00:00:00 -0300

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, Daniel Paşca — Wed, 20 Mar 2024 00:00:00 -0300

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 — Fri, 05 Apr 2024 00:00:00 -0300

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 — Wed, 10 Apr 2024 00:00:00 -0300

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 $R$.

Distance Laplacian eigenvalues of graphs, and chromatic and independence number

Shariefuddin Pirzada, Saleem Khan — Tue, 16 Apr 2024 00:00:00 -0300

Given an interval $I$, let $m_{D^{L} (G)} I$ (or simply $m_{D^{L}} I$) be the number of distance Laplacian eigenvalues of a graph $G$ which lie in $I$. For a prescribed interval $I$, we give the bounds for $m_{D^{L} }I$ in terms of the independence number $\alpha(G)$, the chromatic number $\chi$, the number of pendant vertices $p$, the number of components in the complement graph $C_{\overline{G}}$ and the diameter $d$ of $G$. In particular, we prove that $m_{D^{L}(G) }[n,n+2)\leq \chi-1$, $m_{D^{L}(G)}[n,n+\alpha(G))\leq n-\alpha(G)$, $m_{D^{L}(G) }[n,n+p)\leq n-p$ and discuss the cases where the bounds are best possible. In addition, we characterize graphs of diameter $d\leq 2$ which satisfy $m_{D^{L}(G) } (2n-1,2n )= \alpha(G)-1=\frac{n}{2}-1$. We also propose some problems of interest.

Limit behaviors for a $\beta$-mixing sequence in the St. Petersburg game

Yu Miao, Qing Yin, Zhen Wang — Wed, 24 Apr 2024 00:00:00 -0300

We consider a sequence of non-negative $\beta$-mixing random variables $\{X,X_n : n\geq1\}$ from the classical St. Petersburg game. The accumulated gains $S_n=X_1+X_2+\cdots+X_n$ in the St. Petersburg game are studied, and the large deviations and the weak law of large numbers of $S_n$ are obtained.

Trivial extensions of monomial algebras

María Andrea Gatica, María Valeria Hernández, María Inés Platzeck — Wed, 24 Apr 2024 00:00:00 -0300

We describe the ideal of relations for the trivial extension $T(\Lambda)$ of a finite-dimensional monomial algebra $\Lambda$. When $\Lambda$ is, moreover, a gentle algebra, we solve the converse problem: given an algebra $B$, determine whether $B$ is the trivial extension of a gentle algebra. We characterize such algebras $B$ through properties of the cycles of their quiver, and show how to obtain all gentle algebras $\Lambda$ such that $T(\Lambda) \cong B$. We prove that indecomposable trivial extensions of gentle algebras coincide with Brauer graph algebras with multiplicity one in all vertices in the associated Brauer graph, result proven by S. Schroll.

On the Eneström–Kakeya theorem and its various forms in the quaternionic setting

Abdullah Mir — Wed, 24 Apr 2024 00:00:00 -0300

We study the extensions of the classical Eneström–Kakeya theorem and its various generalizations regarding the distribution of zeros of polynomials from the complex to the quaternionic setting. We aim to build upon the previous work by various authors and derive zero-free regions of some special regular functions of a quaternionic variable with restricted coefficients, namely quaternionic coefficients whose real and imaginary components or moduli of the coefficients satisfy suitable inequalities. The obtained results for this subclass of polynomials and slice regular functions produce generalizations of a number of results known in the literature on this subject.