Published volumes
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 62, number 1 (2021)
June 2021
Front matter

Orthogonality of the Dickson
polynomials of the $(k+1)$th kind.
Diego Dominici
We study the Dickson polynomials of the $(k+1)$th kind over the field of
complex numbers. We show that they are a family of corecursive orthogonal
polynomials with respect to a quasidefinite moment functional $L_{k}$. We
find an integral representation for $L_{k}$ and compute explicit expressions
for all of its moments.

1–30 
Timefrequency analysis associated
with the Laguerre wavelet transform.
Hatem Mejjaoli and Khalifa Trimèche
We define the localization operators associated with Laguerre wavelet
transforms. Next, we prove the boundedness and compactness of these operators,
which depend on a symbol and two admissible wavelets on
$L^{p}_{\alpha}(\mathbb{K})$, $1 \leq p \leq \infty$.

31–55 
Uniform
approximation of Muckenhoupt weights on fractals by simple functions.
Marilina Carena and Marisa Toschi
Given an $A_p$Muckenhoupt weight on a fractal obtained as the attractor of an
iterated function system, we construct a sequence of approximating weights,
which are simple functions belonging uniformly to the $A_p$ class on the
approximating spaces.

57–66 
Gotzmann monomials in four variables.
Vittoria Bonanzinga and Shalom Eliahou
It is a widely open problem to determine which monomials in
the $n$variable polynomial ring $K[x_1,\dots,x_n]$ over a field $K$ have the
Gotzmann property, i.e. induce a Borelstable Gotzmann monomial ideal.
Since 2007, only the case $n \le 3$ was known. Here we solve the problem for
the case $n=4$. The solution involves a surprisingly intricate characterization.

67–93 
Signed graphs with totally disconnected star complements.
Zoran Stanić
We are interested in a signed graph $\dot{G}$ which admits a
decomposition into a totally disconnected (i.e., without edges) star
complement and a signed graph $\dot{S}$ induced by the star set. In this study
we derive certain properties of $\dot{G}$; for example, we prove that the
number of (distinct) eigenvalues of $\dot{S}$ does not exceed the number of
those of $\dot{G}$. Some particular cases are also considered.

95–104 
The weakly zerodivisor graph of a commutative ring.
Mohammad Javad Nikmehr, Abdolreza Azadi, and Reza Nikandish
Let $R$ be a commutative ring with identity, and let $Z(R)$ be the set of
zerodivisors of $R$. The weakly zerodivisor graph of $R$ is the undirected
(simple) graph $W\Gamma(R)$ with vertex set $Z(R)^*$, and two distinct
vertices $x$ and $y$ are adjacent if and only if there exist $r\in \operatorname{ann}(x)$ and
$s\in \operatorname{ann}(y)$ such that $rs=0$. It follows that $W\Gamma(R)$ contains the
zerodivisor graph $\Gamma(R)$ as a subgraph. In this paper, the
connectedness, diameter, and girth of $W\Gamma(R)$ are investigated. Moreover,
we determine all rings whose weakly zerodivisor graphs are star.
We also give conditions under which weakly zerodivisor and
zerodivisor graphs are identical. Finally, the chromatic number of
$W\Gamma(R)$ is studied.

105–116 
A characterization of the Lorentz space
$L(p,r)$ in terms of Orlicz type classes.
Calixto P. Calderón and Alberto Torchinsky
We describe the Lorentz space $L(p,r)$, $0 < r < p$, $p > 1$, in terms of Orlicz type
classes of functions $L_{\Psi}$. As a consequence of this result it follows
that Stein's characterization of the real functions on $\mathbb{R}^n$ that are
differentiable at almost all the points in $\mathbb{R}^n$
[Ann. of Math 113 (1981), no. 2, 383–385],
is equivalent
to the characterization of those functions given by A. P. Calderón
[Riv. Mat. Univ. Parma 2 (1951), 203–213].

117–122 
Classical simple Lie 2algebras
of odd toral rank and a contragredient Lie 2algebra of toral rank 4.
Carlos R. Payares Guevara and Fabián A. Arias Amaya
After the classification of simple Lie algebras over a field of characteristic
$p > 3$, the main problem not yet solved in the theory of finite dimensional
Lie algebras is the classification of simple Lie algebras over a field of
characteristic 2. The first result for this classification problem ensures
that all finite dimensional Lie algebras of absolute toral rank 1 over an
algebraically closed field of characteristic 2 are soluble. Describing simple
Lie algebras (respectively, Lie $2$algebras) of finite dimension of absolute
toral rank (respectively, toral rank) 3 over an algebraically
closed field of characteristic 2 is still an open problem.
In this paper we show that there are no classical type simple Lie $2$algebras with
toral rank odd and furthermore that the simple contragredient Lie $2$algebra
$G(F_{4, a})$ of dimension 34 has toral rank 4. Additionally, we give
the Cartan decomposition of $G(F_{4, a})$.

123–139 
On commutative homogeneous vector bundles attached to nilmanifolds.
Rocío Díaz Martín and Linda Saal
The notion of Gelfand pair $(G,K)$ can be generalized by considering
homogeneous vector bundles over $G/K$ instead of the homogeneous space $G/K$
and matrixvalued functions instead of scalarvalued functions. This gives the
definition of commutative homogeneous vector bundles. Being a Gelfand pair is
a necessary condition for being a commutative homogeneous vector bundle. In
the case when $G/K$ is a nilmanifold having squareintegrable representations,
a big family of commutative homogeneous vector bundles was determined in
[Transform. Groups 24 (2019), no. 3, 887–911].
In this paper we complete that classification.

141–151 
Structure of simple multiplicative HomJordan algebras.
Chenrui Yao, Yao Ma, and Liangyun Chen
We study the structure of simple multiplicative HomJordan
algebras. We discuss equivalent conditions for multiplicative HomJordan
algebras to be solvable, simple, and semisimple. Moreover, we
give a theorem on the classification of simple multiplicative HomJordan
algebras and obtain some propositions about bimodules of multiplicative
HomJordan algebras.

153–170 
The fibering map approach for a singular elliptic
system involving the $p(x)$Laplacian and nonlinear boundary conditions.
Mouna Kratou and Kamel Saoudi
The purpose of this work is to study the existence and multiplicity of
positive solutions for a class of singular elliptic systems involving
the $p(x)$Laplace operator and nonlinear boundary conditions.

171–189 
Cofiniteness of local cohomology modules
in the class of modules in dimension less than a fixed integer.
Alireza Vahidi and Mahdieh PapariZarei
Let $n$ be a nonnegative integer, $R$ a commutative Noetherian ring with
$\dim(R)\leq n+ 2$, $\mathfrak{a}$ an ideal of $R$, and $X$ an arbitrary
$R$module. In this paper, we first prove that $X$ is an
$(\operatorname{FD}_{< n}, \mathfrak{a})$cofinite $R$module if $X$ is an
$\mathfrak{a}$torsion $R$module such that
$\operatorname{Hom}_{R}\big(\frac{R}{\mathfrak{a}}, X\big)$ and
$\operatorname{Ext}_{R}^{1}\big(\frac{R}{\mathfrak{a}}, X\big)$ are
$\operatorname{FD}_{< n}$ $R$modules. Then, we show that
$\operatorname{H}^{i}_{\mathfrak{a}}(X)$ is an $(\operatorname{FD}_{< n},
\mathfrak{a})$cofinite $R$module and $\{\mathfrak{p}\in
\operatorname{Ass}_R(\operatorname{H}^{i}_{\mathfrak{a}}(X)) :
\dim\big(\frac{R}{\mathfrak{p}}\big)\geq n\}$ is a finite set for all $i$ when
$\operatorname{Ext}_{R}^{i}\big(\frac{R}{\mathfrak{a}}, X\big)$ is an
$\operatorname{FD}_{< n}$ $R$module for all $i\leq n+ 2$. As a consequence,
it follows that $\operatorname{Ass}_R(\operatorname{H}^{i}_{\mathfrak{a}}(X))$
is a finite set for all $i$ whenever $R$ is a semilocal ring with
$\dim(R)\leq 3$ and $X$ is an $\operatorname{FD}_{< 1}$ $R$module. Finally,
we observe that the category of $(\operatorname{FD}_{< n},
\mathfrak{a})$cofinite $R$modules forms an Abelian subcategory of the
category of $R$modules.

191–198 
Canal surfaces with generalized 1type Gauss map.
Jinhua Qian, Mengfei Su, and Young Ho Kim
This work considers a kind of classification of canal surfaces in terms of
their Gauss map $\mathbb{G}$ in Euclidean 3space. We introduce the notion of
generalized 1type Gauss map for a submanifold that satisfies $\Delta
\mathbb{G}=f\mathbb{G}+gC$, where $\Delta$ is the Laplace operator, $C$ is a
constant vector, and $(f, g)$ are nonzero smooth functions. First of all, we
show that the Gauss map of any surface of revolution with unit speed profile
curve in Euclidean 3space is of generalized 1type. At the same time, the
canal surfaces with generalized 1type Gauss map are discussed.

199–211 
Selberg zetafunction associated to compact Riemann surface is prime.
Ramūnas Garunkštis
Let $Z(s)$ be the Selberg zetafunction associated to
a compact Riemann surface. We consider decompositions $Z(s)=f(h(s))$, where
$f$ and $h$ are meromorphic functions, and show that such decompositions can
only be trivial.

213–218 
Weight distribution of cyclic codes defined by quadratic forms and related curves.
Ricardo A. Podestá and Denis E. Videla
We consider cyclic codes $\mathcal{C}_\mathcal{L}$ associated to quadratic
trace forms in $m$ variables $Q_R(x) = \operatorname{Tr}_{q^m/q}(xR(x))$
determined by a family $\mathcal{L}$ of $q$linearized polynomials $R$
over $\mathbb{F}_{q^m}$, and three related codes $\mathcal{C}_{\mathcal{L},0}$,
$\mathcal{C}_{\mathcal{L},1}$, and $\mathcal{C}_{\mathcal{L},2}$. We describe
the spectra for all these codes when $\mathcal{L}$ is an even rank family, in
terms of the distribution of ranks of the forms $Q_R$ in the
family $\mathcal{L}$, and we also compute the complete weight enumerator
for $\mathcal{C}_\mathcal{L}$. In particular, considering the family
$\mathcal{L} = \langle x^{q^\ell} \rangle$, with $\ell$ fixed in $\mathbb{N}$,
we give the weight distribution of four parametrized families of cyclic codes
$\mathcal{C}_\ell$, $\mathcal{C}_{\ell,0}$, $\mathcal{C}_{\ell,1}$, and
$\mathcal{C}_{\ell,2}$ over $\mathbb{F}_q$ with zeros $\{ \alpha^{(q^\ell+1)}
\}$, $\{ 1, \alpha^{(q^\ell+1)} \}$, $\{ \alpha^{1},\alpha^{(q^\ell+1)} \}$,
and $\{ 1,\alpha^{1},\alpha^{(q^\ell+1)}\}$ respectively, where $q = p^s$
with $p$ prime, $\alpha$ is a generator of $\mathbb{F}_{q^m}^*$, and
$m/(m,\ell)$ is even. Finally, we give simple necessary and sufficient
conditions for Artin–Schreier curves $y^py = xR(x) + \beta x$, $p$ prime,
associated to polynomials $R \in \mathcal{L}$ to be optimal. We then obtain
several maximal and minimal such curves in the case $\mathcal{L} = \langle
x^{p^\ell}\rangle$ and $\mathcal{L} = \langle x^{p^\ell}, x^{p^{3\ell}}
\rangle$.

219–242 
Recurrent curvature over fourdimensional homogeneous manifolds.
Milad Bastami, Ali HajiBadali, and Amirhesam Zaeim
Recurrent curvature properties are considered on fourdimensional
pseudoRiemannian homogeneous manifolds with nontrivial isotropy, and also on
some geometric manifolds.

243–255 
On Egyptian fractions of length 3.
Cyril Banderier, Carlos Alexis Gómez Ruiz, Florian Luca, Francesco Pappalardi, and Enrique Treviño
Let $a,n$ be positive integers that are relatively prime.
We say that $a/n$ can be represented as an Egyptian fraction of length $k$ if
there exist positive integers $m_1, \ldots, m_k$
such that $\frac{a}{n} = \frac{1}{m_1}+ \cdots + \frac{1}{m_k}$.
Let $A_k(n)$ be the number of solutions $a$ to this equation.
In this article, we give a formula for $A_2(p)$ and
a parametrization for Egyptian fractions of length $3$,
which allows us to give bounds to $A_3(n)$,
to $f_a(n) = \#\{(m_1,m_2,m_3) : \frac{a}{n} =
\frac{1}{m_1}+\frac{1}{m_2}+\frac{1}{m_3}\}$, and finally to $F(n) =
\#\{(a,m_1,m_2,m_3) : \frac{a}{n}=\frac{1}{m_1}+\frac{1}{m_2}+\frac{1}{m_3}\}$.

257–274 
