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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 61, number 1 (2020)
June 2020
Front matter

Local solvability
of elliptic equations of even order with Hölder coefficients.
María Amelia Muschietti and Federico Tournier
We consider elliptic equations of order $2m$ with Hölder coefficients. We
show local solvability of the Dirichlet problem with $m$ conditions on the
boundary of the upper half space. First we consider local solvability in free
space and then we treat the boundary case. Our method is based on applying the
operator to an approximate solution and iterating in the Hölder spaces. A
priori estimates for the approximate solution is the essential part of the
paper.

1–47 
On convergence
of subspaces generated by dilations of polynomials. An application to best
local approximation.
Fabián E. Levis and Claudia V. Ridolfi
We study the convergence of a net of subspaces generated by dilations of
polynomials in a finite dimensional subspace. As a consequence, we extend the
results given by Zó and Cuenya [Advanced Courses of Mathematical Analysis
II (Granada, 2004), 193–213, World Scientific, 2007] on a general
approach to the problems of best vectorvalued approximation on small regions
from a finite dimensional subspace of polynomials.

49–62 
Perturbation of Ruelle resonances and
Faure–Sjöstrand anisotropic space.
Yannick Guedes Bonthonneau
Given an Anosov vector field $X_0$, all sufficiently close vector fields are
also of Anosov type. In this note, we check that the anisotropic spaces
described by Faure and Sjöstrand and by Dyatlov and Zworski can be chosen
adapted to any smooth vector field sufficiently close to $X_0$ in $C^1$ norm.

63–72 
Generalized
metallic structures.
Adara M. Blaga and Antonella Nannicini
We study the properties of a generalized metallic, a generalized product and a
generalized complex structure induced on the generalized tangent bundle of a
smooth manifold $M$ by a metallic Riemannian structure $(J,g)$ on $M$,
providing conditions for their integrability with respect to a suitable
connection. Moreover, by using methods of generalized geometry, we lift $(J,g)$
to metallic Riemannian structures on the tangent and cotangent bundles of $M$,
underlying the relations between them.

73–86 
A heat conduction
problem with sources depending on the average of the heat flux on the
boundary.
Mahdi Boukrouche and Domingo A. Tarzia
Motivated by the modeling of temperature regulation in some mediums, we
consider the nonclassical heat conduction equation in the domain
$D=\mathbb{R}^{n1}\times\mathbb{R}^{+}$ for which the internal energy supply
depends on an average in the time variable of the heat flux $(y, s)\mapsto
V(y,s)= u_{x}(0, y, s)$ on the boundary $S=\partial D$. The solution to the
problem is found for an integral representation depending on the heat flux on
$S$ which is an additional unknown of the considered problem. We obtain that
the heat flux $V$ must satisfy a Volterra integral equation of the second kind
in the time variable $t$ with a parameter in $\mathbb{R}^{n1}$. Under some
conditions on data, we show that a unique local solution exists, which can be
extended globally in time. Finally in the onedimensional case, we obtain the
explicit solution by using the Laplace transform and the Adomian decomposition
method.

87–101 
On classes of finite rings.
Aleksandr Tsarev
A class of rings is a formation whenever it contains all homomorphic images of
its members and if it is subdirect product closed. In the present paper, it is
shown that the lattice of all formations of finite rings is algebraic and
modular. Let $R$ be a finite commutative ring with an identity element. It is
established that there is a onetoone correspondence between the set of all
invariant fuzzy prime ideals of $R$ and the set of all fuzzy prime ideals of
each ring of the formation generated by $R$.

103–111 
A canonical distribution on isoparametric submanifolds I.
Cristián U. Sánchez
We show that on every compact, connected homogeneous isoparametric
submanifold $M$ of codimension $h\geq 2$ in a Euclidean space, there
exists a canonical distribution which is bracket generating of step 2. An
interesting consequence of this fact is also indicated. In this first part
we consider only the case in which the system of restricted roots is
reduced, reserving for a second part the case of nonreduced restricted
roots.

113–130 
Second cohomology space
of $\mathfrak{sl}(2)$ acting on the space of bilinear bidifferential
operators.
Imed Basdouri, Sarra Hammami, and Olfa Messaoud
We consider the $\mathfrak{sl}(2)$module structure on the spaces
of bilinear bidifferential operators acting on the spaces of weighted
densities.
We compute the second cohomology group of the Lie algebra
$\mathfrak{sl}(2)$ with coefficients in the space of bilinear
bidifferential operators that act on tensor densities
$\mathcal{D}_{\lambda,\nu,\mu}$.

131–143 
Kreĭn space unitary dilations
of Hilbert space holomorphic semigroups.
Stefania A. M. Marcantognini
The infinitesimal generator $A$ of a strongly continuous semigroup on a Hilbert
space is assumed to satisfy that $B_\beta:=A\beta$ is a sectorial operator of
angle less than $\frac{\pi}{2}$ for some $\beta \geq 0$. If $B_\beta$ is
dissipative in some equivalent scalar product then the Naimark–Arocena
representation theorem is applied to obtain a Kreĭn space unitary dilation
of the semigroup.

145–160 
Lifting vector fields from manifolds
to $r$jet prolongation of the tangent bundle.
Jan Kurek and Włodzimierz M. Mikulski
If $m\geq 3$ and $r\geq 0$, we deduce that any natural linear operator lifting
vector fields from an $m$manifold $M$ to the $r$jet prolongation $J^{r}TM$ of the
tangent bundle $TM$ is the composition of the flow lifting $\mathcal{J}^r$
corresponding to the $r$jet prolongation functor $J^r$ with a natural linear
operator lifting vector fields from $M$ to $TM$. If $0\leq s\leq
r$ and $m\geq 3$, we find all natural linear operators transforming vector
fields on $M$ into basepreserving fibred maps $J^{r}TM\to J^{s}TM$.

161–168 
Localization operators and
scalogram associated with the generalized continuous wavelet transform on
$\mathbb{R}^d$ for the Heckman–Opdam theory.
Hatem Mejjaoli and Khalifa Trimèche
We consider the generalized wavelet transform $\Phi_{h}^{W}$ on $\mathbb{R}^{d}$ for
the Heckman–Opdam theory. We study the localization operators associated with
$\Phi_{h}^{W}$; in particular, we prove that they are in the Schatten–von
Neumann class. Next we introduce some results on the scalogram for this transform.

169–196 
Volume 61, number 2 (2020)
December 2020
Front matter

$\mathfrak{D}^\perp$invariant real hypersurfaces in complex Grassmannians of rank two.
RuennHuah Lee and TeeHow Loo
Let $M$ be a real hypersurface in a complex Grassmannian of rank
two. Denote by $\mathfrak{J}$ the quaternionic Kähler structure
of the ambient space, $TM^\perp$ the normal bundle over $M$, and
$\mathfrak{D}^\perp=\mathfrak{J}TM^\perp$. The real hypersurface
$M$ is said to be $\mathfrak{D}^\perp$invariant if
$\mathfrak{D}^\perp$ is invariant under the shape operator of $M$.
We show that if $M$ is $\mathfrak{D}^\perp$invariant, then $M$ is
Hopf. This improves the results of Berndt and Suh [Int. J.
Math. 23 (2012) 1250103] and [Monatsh. Math.
127 (1999), 1–14]. We also classify $\mathfrak{D}^\perp$
real hypersurfaces in complex Grassmannians of rank two of
noncompact type with constant principal curvatures.

197–207 
Drazin inverse
of singular adjacency matrices of directed weighted
cycles.
Andrés M. Encinas, Daniel A. Jaume, Cristian Panelo, and Adrián Pastine
We present a necessary and sufficient condition for the singularity of
circulant matrices associated with directed weighted cycles. This
condition is simple and independent of the order of matrices from a
complexity point of view. We give explicit and simple formulas for the
Drazin inverse of these circulant matrices. We also provide a
Bjerhammartype condition for the Drazin inverse.

209–227 
On the atomic
and molecular decomposition of weighted Hardy spaces.
Pablo Rocha
The purpose of this article is to give another molecular decomposition for
members of weighted Hardy spaces, different from that given
by Lee and Lin
[J. Funct. Anal. 188 (2002), no. 2, 442–460],
and to review some overlooked details. As an application of this decomposition,
we obtain the boundedness on $H^{p}_{w}(\mathbb{R}^{n})$ of every bounded
linear operator on some $L^{p_0}(\mathbb{R}^n)$ with $1 < p_0 < +\infty$, for
all weights $w \in \mathcal{A}_{\infty}$ and all $0 < p \leq 1$ if $1 <
\frac{r_w 1}{r_w} p_0$, or all $0 < p < \frac{r_w 1}{r_w} p_0$ if $\frac{r_w
1}{r_w} p_0 \leq 1$, where $r_w$ is the critical index of $w$ for the reverse
Hölder condition. In particular, the wellknown results about boundedness of
singular integrals from $H^{p}_w(\mathbb{R}^{n})$ into
$L^{p}_{w}(\mathbb{R}^{n})$ and on $H^{p}_{w}(\mathbb{R}^{n})$ for all $w \in
\mathcal{A}_{\infty}$ and all $0 < p \leq 1$ are established. We also obtain the
$H^{p}_{w^{p}}(\mathbb{R}^{n})$$H^{q}_{w^{q}}(\mathbb{R}^{n})$ boundedness of
the Riesz potential $I_{\alpha}$ for $0 < p \leq 1$, $\frac{1}{q}=\frac{1}{p} 
\frac{\alpha}{n}$, and certain weights $w$.

229–247 
On some fixed
point theorems in abstract duality pairs.
Kinga Cichoń, Mieczysław Cichoń, and Mohamed M. A.
Metwali
We study the existence of fixed points for some functional problems. We focus
on bilinear operators which are extensions of the product of operators
(Banaś–Dhagetype results) and obtain some results on abstract duality pairs
(abstract triples). However, we do not restrict ourselves to these cases; in
fact, some special cases which do no fit in the above description are
indicated. In particular, our approach allows us to study some differential or
integral problems, for which the operators are not acting on a fixed Banach
space.

249–266 
On Jacobson's
lemma and Cline's formula for Drazin inverses.
Dijana Mosić
Under new conditions $bac=bdb$ and $cdb=cac$, we present extensions of
Jacobson's lemma and Cline's formula for the generalized Drazin inverse and
pseudo Drazin inverse in a ring. Applying these results, we give Jacobson's
lemma for the Drazin inverse, group inverse, and ordinary inverse, and Cline's
formula for the Drazin inverse.

267–276 
Reflexivity of
rings via nilpotent elements.
Abdullah Harmanci, Handan Kose, Yosum Kurtulmaz, and
Burcu Ungor
An ideal $I$ of a ring $R$ is called left Nreflexive if for any $a\in
\operatorname{nil}(R)$ and $b\in R$, $aRb \subseteq I$ implies $bRa
\subseteq I$, where $\operatorname{nil}(R)$ is the set of all nilpotent elements
of $R$. The ring $R$ is called left Nreflexive if the zero ideal is
left Nreflexive. We study the properties of left Nreflexive rings and related
concepts. Since reflexive rings and reduced rings are left Nreflexive rings,
we investigate the sufficient conditions for left Nreflexive rings to be
reflexive and reduced. We first consider basic extensions of left Nreflexive
rings. For an idealsymmetric ideal $I$ of a ring $R$, $R/I$ is left
Nreflexive. If an ideal $I$ of a ring $R$ is reduced as a ring without
identity and $R/I$ is left Nreflexive, then $R$ is left Nreflexive. If $R$ is
a quasiArmendariz ring and the coefficients of any nilpotent polynomial in
$R[x]$ are nilpotent in $R$, it is proved that $R$ is left Nreflexive if and
only if $R[x]$ is left Nreflexive. We show that the concept of left
Nreflexivity is weaker than that of reflexivity and stronger than that of
right idempotent reflexivity.

277–290 
$p$Dual frames
and $p$Riesz sequences in quasinormed spaces.
José Alfonso López Nicolás
The present contribution is aimed at obtaining new results in
duality between $p$dual frames and $p$Riesz sequences in quasinormed spaces
with a normalized Schauder basis. We obtain two results which show the
relationship of duality between these concepts. We split a quasinormed space
into a topological sum of two subspaces and use the Schauder basis to
establish a relationship between the $p$dual frames of one of these subspaces
and the $p$Riesz sequences of the dual of the other one. In fact, the main
results are stated in a lightly more general context.

291–301 
Hyponormality of Toeplitz
operators on the Bergman space of an annulus.
Houcine Sadraoui and Mohammed Guediri
A bounded operator $S$ on a Hilbert space is hyponormal if
$S^{\ast}SSS^{\ast}$ is positive. In this work we find necessary conditions
for the hyponormality of the Toeplitz operator $T_{f+\overline{g}}$ on the
Bergman space of the annulus $\{1/2<z<1\}$, where $f$ and $g$ are analytic
and $f$ satisfies a smoothness condition.

303–313 
The geodesic flow on nilmanifolds associated to graphs.
Gabriela P. Ovando
We study the geodesic flow on nilmanifolds associated to graphs. We are
interested in the construction of first integrals to show complete
integrability on some compact quotients. We start on the corresponding Lie
group equipped with a leftinvariant metric, which is induced to the quotients.
Also examples of integrable geodesic flows and of nonintegrable ones are shown.

315–338 
Quasimodal operators on distributive nearlattices.
Ismael Calomino, Sergio A. Celani, and Luciano J. González
We introduce the notion of quasimodal operator in the variety of distributive
nearlattices, which turns out to be a generalization of the necessity modal
operator studied in [S. Celani and I. Calomino, Math. Slovaca 69
(2019), no. 1, 35–52]. We show that there is a one to one correspondence between a
particular class of quasimodal operators on a distributive nearlattice and
the class of possibility modal operators on the distributive lattice of its
finitely generated filters. Finally, we consider the concept of quasimodal
congruence, and we show that the lattice of quasimodal congruences of a
quasimodal distributive nearlattice is isomorphic to the lattice of
congruences of the lattice of finitely generated filters with a possibility
modal operator.

339–352 
Geometry of pointwise CRslant
warped products in Kaehler manifolds.
BangYen Chen, Siraj Uddin, and Falleh R. AlSolamy
We call a submanifold $M$ of a Kaehler manifold $\tilde{M}$ a pointwise
CRslant warped product if it is a warped product, $B\times_{\!f} N_\theta$, of a
CRproduct $B=N_T\times N_\perp$ and a proper pointwise slant submanifold
$N_\theta$ with slant function $\theta$, where $N_T$ and $N_\perp$ are complex
and totally real submanifolds of $\tilde{M}$. We prove that if a pointwise
CRslant warped product $B\times_{\!f} N_\theta$ with $B=N_T\times N_\perp$ in a
Kaehler manifold is weakly ${\mathfrak{D}^\theta}$totally geodesic, then it
satisfies
\[
\\sigma\^2\geq 4s\left\{ (\csc^2\theta+\cot^2\theta)\\nabla^T(\ln f)\^2 +
(\cot^2\theta)\\nabla^\bot(\ln f)\^2 \right\},
\]
where $N_T$, $N_\perp$, and $N_\theta$ are complex, totally real and proper
pointwise slant submanifolds of $\tilde{M}$, respectively, and
$s=\frac{1}{2}\dim N_\theta$. In this paper we also investigate the equality
case of the inequality. Moreover, we give a nontrivial example and provide
some applications of this inequality.

353–365 
Triangular spherical dihedral ftilings:
the $(\pi/2, \pi/3, \pi/4)$ and $(2\pi/3, \pi/4, \pi/4)$ family.
Catarina P. Avelino and Altino F. Santos
We classify all the dihedral ftilings with spherical triangles
$\big(\frac{\pi}{2}, \frac{\pi}{3}, \frac{\pi}{4}\big)$ and
$\big(\frac{2\pi}{3}, \frac{\pi}{4}, \frac{\pi}{4}\big)$, and give the
combinatorial structure, including the symmetry group of each tiling.

367–387 
Viscosity approximation method for
modified split generalized equilibrium and fixed point problems.
Hammed Anuoluwapo Abass, Chinedu Izuchukwu, and Oluwatosin Temitope Mewomo
We introduce a viscosity iterative algorithm for
approximating a common solution of a modified split generalized equilibrium
problem and a fixed point problem for a quasipseudocontractive mapping which also
solves some variational inequality problems in real Hilbert spaces. The
proposed iterative algorithm is constructed in such a way that it does not require
the prior knowledge of the operator norm. Furthermore, we
prove a strong convergence theorem for approximating the common solution of the
aforementioned problems. Finally, we give a numerical example of our main
theorem. Our result complements and extends some related works in the literature.

389–411 
Minimal solutions of the
rational interpolation problem.
Teresa Cortadellas Benítez, Carlos D'Andrea, and Eulàlia Montoro
We explore connections between the approach of solving the rational
interpolation problem via resolutions of ideals and syzygies, and the standard
method provided by the Extended Euclidean Algorithm (EEA). As a consequence, we
obtain explicit descriptions for solutions of minimal degrees in terms of
the degrees of elements appearing in the EEA. This result allows us to describe the
minimal degree in a $\mu$basis of a polynomial planar parametrization in
terms of a critical degree arising in the EEA.

413–429 
On the restricted partition
function via determinants with Bernoulli polynomials. II.
Mircea Cimpoeaş
Let $r\geq 1$ be an integer, $\mathbf{a}=(a_1,\dots,a_r)$ a vector of positive
integers, and let $D\geq 1$ be a common multiple of $a_1,\dots,a_r$. We prove
that if $D=1$ or $D$ is a prime number then the restricted partition function
$p_{\mathbf{a}}(n) := $ the number of integer solutions $(x_1,\dots,x_r)$ to $\sum_{j=1}^r
a_j x_j=n$, with $x_1\geq 0, \dots, x_r\geq 0$, can be computed by solving a
system of linear equations with coefficients that are values of Bernoulli
polynomials and Bernoulli–Barnes numbers.

431–440 
Weakly convex and convex
domination numbers for generalized Petersen and flower snark graphs.
Jozef Kratica, Dragan Matić, and Vladimir Filipović
We consider the weakly convex and convex domination numbers for two classes of
graphs: generalized Petersen graphs and flower snark graphs.
For a given generalized Petersen graph $GP(n,k)$, we prove that if $k=1$ and
$n\geq 4$ then both the weakly convex domination number $\gamma_\mathit{wcon}(GP(n,k))$ and
the convex domination number $\gamma_\mathit{con}(GP(n,k))$ are equal to $n$. For $k\geq
2$ and $n\geq 13$, $\gamma_\mathit{wcon}(GP(n,k))=\gamma_\mathit{con}(GP(n,k))=2n$, which
is the order of $GP(n,k)$. Special cases for smaller graphs are solved by the
exact method. For a flower snark graph $J_n$, where $n$ is odd and $n\geq 5$, we prove
that $\gamma_\mathit{wcon}(J_n)=2n$ and $\gamma_\mathit{con}(J_n)=4n$.

441–455 
Characterization of
hypersurface singularities in positive characteristic.
Amir Shehzad, Muhammad Ahsan Binyamin, and Hasan Mahmood
The classification of right unimodal and bimodal hypersurface singularities
over a field of positive characteristic was given by H. D. Nguyen. The
classification is described in the style of Arnold and not in an algorithmic
way. This classification was characterized by
M. A. Binyamin et al. [Bull. Math. Soc. Sci. Math. Roumanie (N.S.)
61(109) (2018), no. 3, 333–343]
for the case when the
corank of hypersurface singularities is $\leq 2$.
The aim of this article is to characterize the right unimodal and bimodal
hypersurface singularities of corank $3$ in an algorithmic way by means of
easily computable invariants such as the multiplicity, the Milnor number of the
given equation, and its blowingup.
On the basis of this characterization we implement an algorithm to compute
the type of the right unimodal and bimodal hypersurface singularities without
computing the normal form in the computer algebra system
Singular.

457–468 
