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19521968 Revista de la Unión Matemática Argentina y de la Asociación Física Argentina
19441951 Revista de la Unión Matemática Argentina; órgano de la Asociación Física Argentina
19361944

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.

1–31 
On the zeros of univariate Epolynomials.
María Laura Barbagallo, Gabriela Jeronimo, and Juan Sabia
We consider two problems concerning real zeros of univariate Epolynomials.
First, we prove an explicit upper bound for the absolute values of the
zeroes of an Epolynomial 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 Epolynomials.
This result gives, in particular, an upper bound for the number of real
zeroes of an Epolynomial. We show this bound is sharp for particular
families of these functions, which proves that a conjecture by D. Richardson is
false.

33–46 
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 nonempty 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.

47–65 
The affine logAleksandrov–Fenchel inequality.
ChangJian Zhao
We establish a new affine logAleksandrov–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 logAleksandrov–Fenchel
inequality. The affine logMinkowski and logAleksandrov–Fenchel
inequalities are also derived.

67–77 
Global
attractors in the parametrized Hénon–Devaney map.
Bladismir Leal and Sergio Muñoz
Given two positive real numbers $a$ and $c$, we consider the (twoparameter)
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 nonbounded closed set with fractal structure which
is a global attractor. Our approach leads to an indepth understanding of
the Hénon–Devaney map and its perturbations.

79–101 
Convolutionfactorable 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.

103–117 
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.

119–133 
On conformal
geometry of fourdimensional generalized symmetric spaces.
Amirhesam Zaeim, Yadollah AryaNejad, and Mokhtar Gheitasi
We study conformal geometry on an important class of fourdimensional
(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.

135–153 
The
Extalgebra 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.

155–164 
On certain
regular nicely distancebalanced graphs.
Blas Fernández, Štefko Miklavič, and Safet Penjić
A connected graph $\Gamma$ is called nicely distancebalanced, 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 distancebalanced graphs
with $\gamma = d$ are precisely complete graphs and cycles of length $2d$
or $2d+1$. In this paper we classify regular nicely distancebalanced
graphs with $\gamma=d+1$.

165–185 
Primegenerating 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 primeproducing quadratic polynomials.

187–196 
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 squarefree 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.

197–211 
Stable quasiperiodic
orbits of a class of quintic Duffing systems.
Homero G. DíazMarín and Osvaldo Osuna
For a Duffingtype oscillator with constant damping, a unique odd
nonlinearity, and timedependent coefficients which are quasiperiodic, we
prove existence and stability conditions of quasiperiodic 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.

213–224 
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 fourdimensional 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.

225–228 
Characterization of essential spectra by quasicompact perturbations.
Faiçal Abdmouleh, Hamadi Chaâben, and Ines Walha
We are interested in the concept of quasicompact 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 semiFredholm, upper and
lower semiWeyl, and upper and lower semiBrowder operators, and to provide a fine
description and characterization of some Browder's essential spectra involving this kind
of operators.

229–244 
The full group of isometries of some compact Lie groups endowed with a biinvariant 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 biinvariant Riemannian
metrics.

245–262 
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 $H$.

263–275 
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 twosided 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.

277–284 
Weighted mixed weaktype inequalities for multilinear fractional operators.
M. Belén Picardi
The aim of this paper is to obtain mixed weaktype 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 vectorvalued weighted mixed inequality for $\mathcal{I}_{\alpha}(\vec{f})$
is provided.

285–294 
Parallel skewsymmetric tensors on 4dimensional metric Lie algebras.
Andrea C. Herrera
We give a complete classification, up to isometric isomorphism and scaling, of
4dimensional metric Lie algebras $(\mathfrak{g},\langle\cdot,\cdot\rangle)$ that admit a
nonzero parallel skewsymmetric 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 skewsymmetric
endomorphism is a multiple of a complex structure.

295–311 
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 nonsingular 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
$G:=V\rtimes\mathbb{R}^n$.

313–329 
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.

331–346 
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.

347–360 
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.

361–374 
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 concentrationbased uncertainty principles, including the
Amrein–Berthier–Benedicks and local inequalities for the deformed Stockwell transform.

375–423 
Duality for infinitedimensional 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
finitedimensional.

425–467 
On Hopf algebras over basic Hopf algebras of dimension 24.
Rongchuan Xiong
We determine finitedimensional Hopf algebras over an algebraically closed field of
characteristic zero, whose Hopf coradical is isomorphic to a nonpointed basic Hopf
algebra of dimension 24 and the infinitesimal braidings are indecomposable objects. In
particular, we obtain families of new finitedimensional Hopf algebras without the dual
Chevalley property.

469–493 
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 coquasitriangular $\ast$bialgebra admit a
symmetrization (or bosonization) and that their Lévy processes are in onetoone
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.

495–532 
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.

533–565 
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 $\lambda>0$.

567–586 
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.

587–609 
