Revista de la
Unión Matemática Argentina

Volume 65, number 1 (2023)

June 2023
Front matter
Invariants of formal pseudodifferential operator algebras and algebraic modular forms. François Dumas and François Martin
We study the question of extending an action of a group $\Gamma$ on a commutative domain $R$ to a formal pseudodifferential operator ring $B=R(\!(x\,;\,d)\!)$ with coefficients in $R$, as well as to some canonical quadratic extension $C=R(\!(x^{1/2}\,;\,\frac 12 d)\!)_2$ of $B$. We give conditions for such an extension to exist and describe under suitable assumptions the invariant subalgebras $B^\Gamma$ and $C^\Gamma$ as Laurent series rings with coefficients in $R^\Gamma$. We apply this general construction to the numbertheoretical context of a subgroup $\Gamma$ of $\mathrm{SL}(2,\mathbb{C})$ acting by homographies on an algebra $R$ of functions in one complex variable. The subalgebra $C_0^\Gamma$ of invariant operators of nonnegative order in $C^\Gamma$ is then linearly isomorphic to the product space $\mathcal{M}_0=\prod_{j\geq 0}M_j$, where $M_j$ is the vector space of algebraic modular forms of weight $j$ in $R$. We obtain a structure of noncommutative algebra on $\mathcal{M}_0$, which can be identified with a space of algebraic Jacobi forms. We study properties of the correspondence $\mathcal{M}_0\to C_0^\Gamma$, whose restriction to even weights was previously known, using arithmetical arguments and the algebraic results of the first part of the article.
On the zeros of univariate E-polynomials. María Laura Barbagallo, Gabriela Jeronimo, and Juan Sabia
We consider two problems concerning real zeros of univariate E-polynomials. First, we prove an explicit upper bound for the absolute values of the zeroes of an E-polynomial defined by polynomials with integer coefficients that improves the bounds known up to now. On the other hand, we extend the classical Budan–Fourier theorem for real polynomials to E-polynomials. This result gives, in particular, an upper bound for the number of real zeroes of an E-polynomial. We show this bound is sharp for particular families of these functions, which proves that a conjecture by D. Richardson is false.
A generalization of the annihilating ideal graph for modules. Soraya Barzegar, Saeed Safaeeyan, and Ehsan Momtahan
We show that an $R$-module $M$ is noetherian (resp., artinian) if and only if its annihilating submodule graph, $\mathbb{G}(M)$, is a non-empty graph and it has ascending chain condition (resp., descending chain condition) on vertices. Moreover, we show that if $\mathbb{G}(M)$ is a locally finite graph, then $M$ is a module of finite length with finitely many maximal submodules. We also derive necessary and sufficient conditions for the annihilating submodule graph of a reduced module to be bipartite (resp., complete bipartite). Finally, we present an algorithm for deriving both $\Gamma (\mathbb{Z}_n)$ and $\mathbb{G}(\mathbb{Z}_n)$ by Maple, simultaneously.
The affine log-Aleksandrov–Fenchel inequality. Chang-Jian Zhao
We establish a new affine log-Aleksandrov–Fenchel inequality for mixed affine quermassintegrals by introducing new concepts of affine and multiple affine measures, and using the newly established Aleksandrov–Fenchel inequality for multiple mixed affine quermassintegrals. Our new inequality yields as special cases the classical Aleksandrov–Fenchel inequality and the $L_{p}$-affine log-Aleksandrov–Fenchel inequality. The affine log-Minkowski and log-Aleksandrov–Fenchel inequalities are also derived.
Global attractors in the parametrized Hénon–Devaney map. Bladismir Leal and Sergio Muñoz
Given two positive real numbers $a$ and $c$, we consider the (two-parameter) family of nonlinear mappings \[ F_{a,c}(x,y)=\left(ax+\frac{1}{y}, \, c y-\frac{c}{y}-a c x\right). \] $F_{1,1}$ is the classical Hénon–Devaney map. For a large region of parameters, we exhibit an invariant non-bounded closed set with fractal structure which is a global attractor. Our approach leads to an in-depth understanding of the Hénon–Devaney map and its perturbations.
Convolution-factorable multilinear operators. Ezgi Erdoğan
We study multilinear operators defined on topological products of Banach algebras of integrable functions and Banach left modules with convolution product. The main theorem of the paper presents a factorization for multilinear operators through convolution that implies the property known as zero product preservation. By using this factorization we investigate properties of multilinear operators including integral representations and we give applications related to orthogonally additive homogeneous polynomials and Hilbert–Schmidt operators.
Ordering of minimal energies in unicyclic signed graphs. Tahir Shamsher, Mushtaq A. Bhat, Shariefuddin Pirzada, and Yilun Shang
Let $S=(G,\sigma)$ be a signed graph of order $n$ and size $m$ and let $t_1,t_2,\dots,t_n$ be the eigenvalues of $S$. The energy of $S$ is defined as $E(S)=\sum_{j=1}^{n}|t_j|$. A connected signed graph is said to be unicyclic if its order and size are the same. In this paper we characterize, up to switching, the unicyclic signed graphs with first $11$ minimal energies for all $n \geq 11$. For $3\leq n \leq 7$, we provide complete orderings of unicyclic signed graphs with respect to energy. For $8 \leq n \leq 10$, we determine unicyclic signed graphs with first $13$ minimal energies.
On conformal geometry of four-dimensional generalized symmetric spaces. Amirhesam Zaeim, Yadollah AryaNejad, and Mokhtar Gheitasi
We study conformal geometry on an important class of four-dimensional (pseudo-)Riemannian manifolds: generalized symmetric spaces. This leads to the general description of conformally Einstein metrics on the spaces under consideration. Finally, the class of oscillator Lie groups is studied for the conformally Einstein property.
The Ext-algebra of the Brauer tree algebra associated to a line. Olivier Dudas
We compute the $\mathrm{Ext}$-algebra of the Brauer tree algebra associated to a line with no exceptional vertex.
On certain regular nicely distance-balanced graphs. Blas Fernández, Štefko Miklavič, and Safet Penjić
A connected graph $\Gamma$ is called nicely distance-balanced, whenever there exists a positive integer $\gamma=\gamma(\Gamma)$ such that, for any two adjacent vertices $u,v$ of $\Gamma$, there are exactly $\gamma$ vertices of $\Gamma$ which are closer to $u$ than to $v$, and exactly $\gamma$ vertices of $\Gamma$ which are closer to $v$ than to $u$. Let $d$ denote the diameter of $\Gamma$. It is known that $d \le \gamma$, and that nicely distance-balanced graphs with $\gamma = d$ are precisely complete graphs and cycles of length $2d$ or $2d+1$. In this paper we classify regular nicely distance-balanced graphs with $\gamma=d+1$.
Prime-generating quadratic polynomials. Víctor Julio Ramírez Viñas
Let $a,b,c$ be integers. We provide a necessary condition for the function $|ax^2 + bx + c|$ to generate only primes for consecutive integers. We then apply this criterion to give sufficient conditions for the real quadratic field $\mathcal{K}=\mathbb{Q}(\sqrt{d})$, $d\in\mathbb{N}$, to have class number one, in terms of prime-producing quadratic polynomials.
On power integral bases of certain pure number fields defined by $x^{84}-m$. Lhoussain El Fadil and Mohamed Faris
Let $K=\mathbb{Q}(\alpha)$ be a pure number field generated by a complex root $\alpha$ of a monic irreducible polynomial $F(x) = x^{84}-m$, with $m\neq\pm 1$ a square-free integer. In this paper, we study the monogeneity of $K$. We prove that if $m \not\equiv 1\pmod{4}$, $m \not\equiv \pm 1\pmod{9}$, and $\overline{m} \not\in \{\pm \overline{1},\pm \overline{18}, \pm \overline{19}\} \pmod{49}$, then $K$ is monogenic. But if $m \equiv 1\pmod{4}$ or $m \equiv\pm 1\pmod{9}$ or $m \equiv 1 \pmod{49}$, then $K$ is not monogenic. Some illustrating examples are given.
Stable quasi-periodic orbits of a class of quintic Duffing systems. Homero G. Díaz-Marín and Osvaldo Osuna
For a Duffing-type oscillator with constant damping, a unique odd nonlinearity, and time-dependent coefficients which are quasi-periodic, we prove existence and stability conditions of quasi-periodic solutions. We thus generalize some results for periodic coefficients and quintic nonlinearity. We use the classical theory of perturbations and present some numerical examples for the quintic case to illustrate our findings.

Volume 65, number 2 (2023)

December 2023
Front matter
An exotic example of a tensor product of a CM elliptic curve and a weight 1 form. Ariel Pacetti
The representation obtained as a tensor product of a rational elliptic curve with a weight 1 modular form is in general an irreducible four-dimensional representation. However, there are some instances where such representation is reducible. In this short article we give an exotic way to obtain such a reducible representation.
Characterization of essential spectra by quasi-compact perturbations. Faiçal Abdmouleh, Hamadi Chaâben, and Ines Walha
We are interested in the concept of quasi-compact operators allowing us to provide some advances on the theory of operators acting in Banach spaces. More precisely, our main objective is to exhibit the importance of the use of this notion to outline a new approach in the analysis of the stability problems of upper and lower semi-Fredholm, upper and lower semi-Weyl, and upper and lower semi-Browder operators, and to provide a fine description and characterization of some Browder's essential spectra involving this kind of operators.
The full group of isometries of some compact Lie groups endowed with a bi-invariant metric. Alberto Dolcetti and Donato Pertici
We describe the full group of isometries of absolutely simple, compact, connected real Lie groups, of $S\mathcal{O}(4)$, and of $U(n)$, endowed with suitable bi-invariant Riemannian metrics.
On maps preserving the Jordan product of $C$-symmetric operators. Zouheir Amara and Mourad Oudghiri
Given a conjugation $C$ on a complex separable Hilbert space $H$, a bounded linear operator $A$ acting on $H$ is said to be $C$-symmetric if $A=CA^*C$. In this paper, we provide a complete description to all those maps on the algebra of linear operators acting on a finite dimensional Hilbert space that preserve the Jordan product of $C$-symmetric operators, in both directions, for every conjugation $C$ on $H$.
Drazin invertibility of linear operators on quaternionic Banach spaces. El Hassan Benabdi and Mohamed Barraa
The paper studies the Drazin inverse for right linear operators on a quaternionic Banach space. Let $A$ be a right linear operator on a two-sided quaternionic Banach space. It is shown that if $A$ is Drazin invertible then the Drazin inverse of $A$ is given by $f(A)$, where $f$ is $0$ in an axially symmetric neighborhood of $0$ and $f(q) = q^{-1}$ in an axially symmetric neighborhood of the nonzero spherical spectrum of $A$. Some results analogous to the ones concerning the Drazin inverse of operators on complex Banach spaces are proved in the quaternionic context.
Weighted mixed weak-type inequalities for multilinear fractional operators. M. Belén Picardi
The aim of this paper is to obtain mixed weak-type inequalities for multilinear fractional operators, extending results by Berra, Carena and Pradolini [J. Math. Anal. Appl. 479 (2019)]. We prove that, under certain conditions on the weights, there exists a constant $C$ such that \[ \Bigg\| \frac{\mathcal{G}_{\alpha}(\vec{f})}{v}\Bigg\|_{L^{q, \infty}(\nu v^q)} \leq C \prod_{i=1}^m{\|f_i\|_{L^1(u_i)}}, \] where $\mathcal{G}_{\alpha}(\vec{f})$ is the multilinear maximal function $\mathcal{M}_{\alpha}(\vec{f})$ introduced by Moen [Collect. Math. 60 (2009)] or the multilineal fractional integral $\mathcal{I}_{\alpha}(\vec{f})$. As an application, a vector-valued weighted mixed inequality for $\mathcal{I}_{\alpha}(\vec{f})$ is provided.
Parallel skew-symmetric tensors on 4-dimensional metric Lie algebras. Andrea C. Herrera
We give a complete classification, up to isometric isomorphism and scaling, of 4-dimensional metric Lie algebras $(\mathfrak{g},\langle\cdot,\cdot\rangle)$ that admit a non-zero parallel skew-symmetric endomorphism. In particular, we distinguish those metric Lie algebras that admit such an endomorphism which is not a multiple of a complex structure, and for each of them we obtain the de Rham decomposition of the associated simply connected Lie group with the corresponding left invariant metric. On the other hand, we find that the associated simply connected Lie group is irreducible as a Riemannian manifold for those metric Lie algebras where each parallel skew-symmetric endomorphism is a multiple of a complex structure.
The cortex of a class of semidirect product exponential Lie groups. Béchir Dali and Chaïma Sayari
In the present paper, we are concerned with the determination of the cortex of semidirect product exponential Lie groups. More precisely, we consider a finite dimensional real vector space $V$ and some abelian matrix group $H=\exp\big(\sum_{i=1}^{n}\mathbb{R} A_i\bigr)$, where ${A_1,\dots, A_n}$ is a set of pairwise commuting non-singular matrices acting on $V$. We first investigate the cortex of the action of the group $H$ on $V$. As an application, we investigate the cortex of the group semidirect product $G:=V\rtimes\mathbb{R}^n$.
The $r$-dynamic edge coloring of a closed helm graph. Raúl M. Falcón, Mathiyazhagan Venkatachalam, Sathasivam Gowri, and Gnanasekaran Nandini
As a natural generalization of the classical coloring problem in graph theory, the dynamic coloring problem deals with the existence of a proper coloring $c$ of a graph so that $|c(N(v))| \geq \min{r, d(v)}$ for every vertex $v$. In this paper, we obtain the $r$-dynamic edge chromatic number of any given closed helm graph for any positive integer $r$. This coincides with the $r$-dynamic chromatic number of the line graph of a closed helm graph.
The John–Nirenberg inequality for Orlicz–Lorentz spaces in a probabilistic setting. Libo Li and Zhiwei Hao
The John–Nirenberg inequality is widely studied in the field of mathematical analysis and probability theory. In this paper we study a new type of the John–Nirenberg inequality for Orlicz–Lorentz spaces in a probabilistic setting. To be precise, let $0 < q \leq \infty$ and $\Phi$ be an $N$-function with some proper restrictions. We prove that if the stochastic basis ${\mathcal{F}_n}_{n \geq 0}$ is regular, then $BMO_{\Phi,q}=BMO$, with equivalent (quasi)-norms. The result is new, which improves previous work on martingale Hardy theory.
The minimal number of homogeneous geodesics depending on the signature of the Killing form. Zdeněk Dušek
The existence of at least two homogeneous geodesics in any homogeneous Finsler manifold was proved in a previous paper by the author. The examples of solvable Lie groups with invariant Finsler metric which admit just two homogeneous geodesics were presented in another paper. In the present work, it is shown that a homogeneous Finsler manifold with indefinite Killing form admits at least four homogeneous geodesics. Examples of invariant Randers metrics on Lie groups with definite Killing form admitting just two homogeneous geodesics and examples with indefinite Killing form admitting just four homogeneous geodesics are presented.
Generalized translation operator and uncertainty principles associated with the deformed Stockwell transform. Hatem Mejjaoli
We study the generalized translation operator associated with the deformed Hankel transform on $\mathbb{R}$. Firstly, we prove the trigonometric form of the generalized translation operator. Next, we derive the positivity of this operator on a suitable space of even functions. Making use of the positivity of the generalized translation operator we introduce and study the deformed Stockwell transform. Knowing the fact that the study of uncertainty principles is both theoretically interesting and practically useful, we formulate several qualitative uncertainty principles for this new integral transform. Firstly, we mainly establish various versions of Heisenberg's uncertainty principles. Secondly, we derive some weighted uncertainty inequalities such as Pitt's and Beckner's uncertainty inequalities for the deformed Stockwell transform. We culminate our study by formulating several concentration-based uncertainty principles, including the Amrein–Berthier–Benedicks and local inequalities for the deformed Stockwell transform.
Duality for infinite-dimensional braided bialgebras and their (co)modules. Elmar Wagner
The paper presents a detailed description of duality for braided algebras, coalgebras, bialgebras, Hopf algebras, and their modules and comodules in the infinite setting. Assuming that the dual objects exist, it is shown how a given braiding induces compatible braidings for the dual objects, and how actions (resp., coactions) can be turned into coactions (resp., actions) of the dual coalgebra (resp., algebra), with an emphasis on braided bialgebras and their braided (co)module algebras. Examples are provided by considering these structures in a graded (or filtered) setting, where each degree is finite-dimensional.
On Hopf algebras over basic Hopf algebras of dimension 24. Rongchuan Xiong
We determine finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero, whose Hopf coradical is isomorphic to a non-pointed basic Hopf algebra of dimension 24 and the infinitesimal braidings are indecomposable objects. In particular, we obtain families of new finite-dimensional Hopf algebras without the dual Chevalley property.
Brownian motion on involutive braided spaces. Uwe Franz, Michael Schürmann, and Monika Varšo
We study (quantum) stochastic processes with independent and stationary increments (i.e., Lévy processes), and in particular Brownian motions in braided monoidal categories. The notion of increments is based on a bialgebra or Hopf algebra structure, and positivity is taken w.r.t. an involution. We show that involutive bialgebras and Hopf algebras in the Yetter–Drinfeld categories of a quasi- or coquasi-triangular $\ast$-bialgebra admit a symmetrization (or bosonization) and that their Lévy processes are in one-to-one correspondence with a certain class of Lévy processes on their symmetrization. We classify Lévy processes with quadratic generators, i.e., Brownian motions, on several braided Hopf-$\ast$-algebras that are generated by their primitive elements (also called braided $\ast$-spaces), and on the braided $\mathrm{SU}(2)$-quantum groups.
Coupling local and nonlocal equations with Neumann boundary conditions. Gabriel Acosta, Francisco Bersetche, and Julio D. Rossi
We introduce two different ways of coupling local and nonlocal equations with Neumann boundary conditions in such a way that the resulting model is naturally associated with an energy functional. For these two models we prove that there is a minimizer of the resulting energy that is unique modulo adding a constant.
Abel ergodic theorems for $\alpha$-times integrated semigroups. Fatih Barki and Abdelaziz Tajmouati
Let ${T(t)}_{t\geq0}$ be an $\alpha$-times integrated semigroup of bounded linear operators on the Banach space $\mathcal{X}$ and let $A$ be their generator. In this paper, we study the uniform convergence of the Abel averages $\mathcal{A} (\lambda) = \lambda^{\alpha +1} \int_{0}^{\infty} e^{-\lambda t} T(t)dt$ as $\lambda \to 0^+$, with $\alpha \geq 0$. More precisely, we show that the following conditions are equivalent: (i) $T(t)$ is uniformly Abel ergodic; (ii) $\mathcal{X}= \mathcal{R}(A)\oplus \mathcal{N}(A)$, with $\mathcal{R}(A)$ closed; (iii) $\|\lambda^2 R(\lambda,A)\| \longrightarrow 0$ as $\lambda\to 0^+$, and $\mathcal{R}(A^k)$ is closed for some integer $k$; (iv) $A$ is $a$-Drazin invertible and $\mathcal{R}(A^k)$ is closed for some $k\geq 1$; where $\mathcal{N}(A)$, $\mathcal{R}(A)$ and $R(\lambda,A)$ are the kernel, the range, and the resolvent function of $A$, respectively. Additionally, we show that if $T(t)$ satisfies $\lim_{t\to \infty}\|T(t)\|/ t^{\alpha+1}=0$, then $T(t)$ is uniformly Abel ergodic if and only if $\frac{1}{t^{\alpha+1}}\int_{0}^{t}T(s)ds$ converges uniformly as $t \to +\infty$. Finally, we examine simultaneously this theory with the uniform power convergence of the Abel averages $\mathcal{A} (\lambda)$ for some $\lambda>0$.
Some inequalities for Lagrangian submanifolds in holomorphic statistical manifolds of constant holomorphic sectional curvature. Yan Jiang, Dandan Cai, and Liang Zhang
We obtain two types of inequalities for Lagrangian submanifolds in holomorphic statistical manifolds of constant holomorphic sectional curvature. One relates the Oprea invariant to the mean curvature, the other relates the Chen invariant to the mean curvature. Our results generalize the corresponding inequalities for Lagrangian submanifolds in complex space forms.