Revista de la
Unión Matemática Argentina

Volume 63, number 1 (2022)

June 2022
Front matter
Spectral distances in some sets of graphs. Irena M. Jovanović
Some of the spectral distance related parameters (cospectrality, spectral eccentricity, and spectral diameter with respect to an arbitrary graph matrix) are determined in one particular set of graphs. According to these results, the spectral distances connected with the adjacency matrix and the corresponding distance related parameters are computed in some sets of trees. Examples are provided of graphs whose spectral distances related to the adjacency matrix, the Laplacian and the signless Laplacian matrix are mutually equal. The conjecture related to the spectral diameter of the set of connected regular graphs with respect to the adjacency matrix is disproved using graph energy.
Approximation via statistical $K_{a}^{2}$-convergence on two-dimensional weighted spaces. Sevda Yıldız
We give a non-regular statistical summability method named statistical $K_{a}^{2}$-convergence and prove a Korovkin type approximation theorem for this new and interesting convergence method on two-dimensional weighted spaces. We also study the rate of statistical $K_{a}^{2}$-convergence by using the weighted modulus of continuity and afterwards we present a non-trivial application.
Some relations between the skew spectrum of an oriented graph and the spectrum of certain closely associated signed graphs. Zoran Stanić
Let $R_{G'}$ be the vertex-edge incidence matrix of an oriented graph $G'$. Let $\Lambda(\dot{F})$ be the signed graph whose vertices are identified as the edges of a signed graph $\dot{F}$, with a pair of vertices being adjacent by a positive (resp. negative) edge if and only if the corresponding edges of $\dot{G}$ are adjacent and have the same (resp. different) sign. In this paper, we prove that $G'$ is bipartite if and only if there exists a signed graph $\dot{F}$ such that $R_{G'}^\intercal R_{G'}-2I$ is the adjacency matrix of $\Lambda(\dot{F})$. It occurs that $\dot{F}$ is fully determined by $G'$. As an application, in some particular cases we express the skew eigenvalues of $G'$ in terms of the eigenvalues of $\dot{F}$. We also establish some upper bounds for the skew spectral radius of $G'$ in both the bipartite and the non-bipartite case.
The convex and weak convex domination number of convex polytopes. Zoran Lj. Maksimović, Aleksandar Lj. Savić, and Milena S. Bogdanović
This paper is devoted to solving the weakly convex dominating set problem and the convex dominating set problem for some classes of planar graphs—convex polytopes. We consider all classes of convex polytopes known from the literature and present exact values of weakly convex and convex domination number for all classes, namely $A_n$, $B_n$, $C_n$, $D_n$, $E_n$, $R_n$, $R''_n$, $Q_n$, $S_n$, $S''_n$, $T_n$, $T''_n$ and $U_n$. When $n$ is up to 26, the values are confirmed by using the exact method, while for greater values of $n$ theoretical proofs are given.
The $\mathfrak{A}$-principal real hypersurfaces in complex quadrics. Tee-How Loo
A real hypersurface in the complex quadric $Q^m=SO_{m+2}/SO_mSO_2$ is said to be $\mathfrak{A}$-principal if its unit normal vector field is singular of type $\mathfrak{A}$-principal everywhere. In this paper, we show that a $\mathfrak{A}$-principal Hopf hypersurface in $Q^m$, $m\geq3$, is an open part of a tube around a totally geodesic $Q^{m+1}$ in $Q^m$. We also show that such real hypersurfaces are the only contact real hypersurfaces in $Q^m$. The classification for complete pseudo-Einstein real hypersurfaces in $Q^m$, $m\geq3$, is also obtained.
On the module intersection graph of ideals of rings. Thangaraj Asir, Arun Kumar, and Alveera Mehdi
Let $R$ be a commutative ring and $M$ an $R$-module. The $M$-intersection graph of ideals of $R$ is an undirected simple graph, denoted by $G_{M}(R)$, whose vertices are non-zero proper ideals of $R$ and two distinct vertices are adjacent if and only if $IM\cap JM\neq 0$. In this article, we focus on how certain graph theoretic parameters of $G_M(R)$ depend on the properties of both $R$ and $M$. Specifically, we derive a necessary and sufficient condition for $R$ and $M$ such that the $M$-intersection graph $G_M(R)$ is either connected or complete. Also, we classify all $R$-modules according to the diameter value of $G_M(R)$. Further, we characterize rings $R$ for which $G_M(R)$ is perfect or Hamiltonian or pancyclic or planar. Moreover, we show that the graph $G_{M}(R)$ is weakly perfect and cograph.
On Baer modules. Chillumuntala Jayaram, Ünsal Tekir, and Suat Koç
A commutative ring $R$ is said to be a Baer ring if for each $a\in R$, $\operatorname{ann}(a)$ is generated by an idempotent element $b\in R$. In this paper, we extend the notion of a Baer ring to modules in terms of weak idempotent elements defined in a previous work by Jayaram and Tekir. Let $R$ be a commutative ring with a nonzero identity and let $M$ be a unital $R$-module. $M$ is said to be a Baer module if for each $m\in M$ there exists a weak idempotent element $e\in R$ such that $\operatorname{ann}_{R}(m)M=eM$. Various examples and properties of Baer modules are given. Also, we characterize a certain class of modules/submodules such as von Neumann regular modules/prime submodules in terms of Baer modules.
A characterization of Stone and linear Heyting algebras. Alejandro Petrovich and Carlos Scirica
An important problem in the variety of Heyting algebras $\mathcal{H}$ is to find new characterizations which allow us to determinate if a given $H\in \mathcal{H}$ is linear or Stone. In this work we present two Heyting algebras, $H^{ns}$ and $H^{snl}$, such that: (a) a Heyting algebra $H$ is a Stone–Heyting algebra if and only if $H^{ns}$ cannot be embedded in $H$, and (b) $H$ is a linear Heyting algebra if and only if neither $H^{ns}$ nor $H^{snl}$ can be embedded in $H$.
Positive periodic solutions of a discrete ratio-dependent predator-prey model with impulsive effects. Cosme Duque and José L. Herrera Diestra
Studies of the dynamics of predator-prey systems are abundant in the literature. In an attempt to account for more realistic models, previous studies have opted for the use of discrete predator-prey systems with ratio-dependent functional response. In the present research we go a step further with the inclusion of impulsive effects in the dynamics. More concretely, by assuming that the coefficients involved in the system and the impulses are periodic, we obtain sufficient conditions for the existence of periodic solutions. We present some numerical examples to illustrate the effectiveness of our results.
The total co-independent domination number of some graph operations. Abel Cabrera Martínez, Suitberto Cabrera García, Iztok Peterin, and Ismael G. Yero
A set $D$ of vertices of a graph $G$ is a total dominating set if every vertex of $G$ is adjacent to at least one vertex of $D$. The total dominating set $D$ is called a total co-independent dominating set if the subgraph induced by $V(G) - D$ is edgeless. The minimum cardinality among all total co-independent dominating sets of $G$ is the total co-independent domination number of $G$. In this article we study the total co-independent domination number of the join, strong, lexicographic, direct and rooted products of graphs.
On the image set and reversibility of shift morphisms over discrete alphabets. Jorge Campos, Neptalí Romero, and Ramón Vivas
We provide sufficient conditions in order to show that the image set of a continuous and shift-commuting map defined on a shift space over an arbitrary discrete alphabet is also a shift space. Additionally, if such a map is injective, then its inverse is also continuous and shift-commuting.
$C$-groups and mixed $C$-groups of bounded linear operators on non-archimedean Banach spaces. Abdelkhalek El Amrani, Aziz Blali, and Jawad Ettayb
We introduce and study $C$-groups and mixed $C$-groups of bounded linear operators on non-archimedean Banach spaces. Our main result extends some existing theorems on this topic. In contrast with the classical setting, the parameter of a given $C$-group (or mixed $C$-group) belongs to a clopen ball $\Omega_{r}$ of the ground field $\mathbb{K}$. As an illustration, we discuss the solvability of some homogeneous $p$-adic differential equations for $C$-groups and inhomogeneous $p$-adic differential equations for mixed $C$-groups when $\alpha=-1$. Examples are given to support our work.
Stability conditions and maximal green sequences in abelian categories. Thomas Brüstle, David Smith, and Hipolito Treffinger
We study the stability functions on abelian categories introduced by Rudakov and their relation with torsion classes and maximal green sequences. Moreover, we introduce the concept of red paths, a stability condition in the sense of Rudakov that captures information of the wall and chamber structure of the category.
Trees with a unique maximum independent set and their linear properties. Daniel A. Jaume, Gonzalo Molina, and Rodrigo Sota
Trees with a unique maximum independent set encode the maxi-mum matching structure in every tree. In this work we study some of their linear properties and give two graph operations, stellare and S-coalescence, which allow building all trees with a unique maximum independent set. The null space structure of any tree can be understood in terms of these graph operations.
New uncertainty principles for the $(k,a)$-generalized wavelet transform. Hatem Mejjaoli
We present the basic $(k,a)$-generalized wavelet theory and prove several Heisenberg-type inequalities for this transform. After reviewing Pitt's and Beckner's inequalities for the $(k,a)$-generalized Fourier transform, we connect both inequalities to show a generalization of uncertainty principles for the $(k,a)$-generalized wavelet transform. We also present two concentration uncertainty principles, namely the Benedicks–Amrein–Berthier's uncertainty principle and local uncertainty principles. Finally, we connect these inequalities to show a generalization of the uncertainty principle of Heisenberg type and we prove the Faris–Price uncertainty principle for the $(k,a)$-generalized wavelet transform.
A Clemens–Schmid type exact sequence over a local basis. Genaro Hernandez-Mada
Let $k$ be a finite field of characteristic $p$ and let $X \rightarrow \operatorname{Spec} k[[t]]$ be a semistable family of varieties over $k$. We prove that there exists a Clemens–Schmid type exact sequence for this family. We do this by constructing a larger family defined over a smooth curve and using a Clemens–Schmid exact sequence in characteristic $p$ for this new family.

Volume 63, number 2 (2022)

December 2022
Front matter
Blow-up of positive-initial-energy solutions for nonlinearly damped semilinear wave equations. Mohamed Amine Kerker
We consider a class of semilinear wave equations with both strongly and nonlinear weakly damped terms, \[ u_{tt}-\Delta u-\omega\Delta u_t+\mu\vert u_t\vert^{m-2}u_t=\vert u\vert^{p-2}u, \] associated with initial and Dirichlet boundary conditions. Under certain conditions, we show that any solution with arbitrarily high positive initial energy blows up in finite time if $m < p$. Furthermore, we obtain a lower bound for the blow-up time.
Binomial edge ideals of cographs. Thomas Kahle and Jonas Krüsemann
We determine the Castelnuovo–Mumford regularity of binomial edge ideals of complement-reducible graphs (cographs). For cographs with $n$ vertices the maximum regularity grows as $2n/3$. We also bound the regularity by graph-theoretic invariants and construct a family of counterexamples to a conjecture of Hibi and Matsuda.
A compact manifold with infinite-dimensional co-invariant cohomology. Mehdi Nabil
Let $M$ be a smooth manifold. When $\Gamma$ is a group acting on $M$ by diffeomorphisms, one can define the $\Gamma$-co-invariant cohomology of $M$ to be the cohomology of the complex $\Omega_c(M)_\Gamma=\operatorname{span}\{\omega-\gamma^*\omega : \omega\in\Omega_c(M),\,\gamma\in\Gamma\}$. For a Lie algebra $\mathcal{G}$ acting on the manifold $M$, one defines the cohomology of $\mathcal{G}$-divergence forms to be the cohomology of the complex $\mathcal{C}_{\mathcal{G}}(M)=\operatorname{span}\{L_X\omega : \omega\in\Omega_c(M),\,X\in\mathcal{G}\}$. In this short paper we present a situation where these two cohomologies are infinite dimensional on a compact manifold.
Conformal vector fields on statistical manifolds. Leila Samereh and Esmaeil Peyghan
Introducing the conformal vector fields on a statistical manifold, we present necessary and sufficient conditions for a vector field on a statistical manifold to be conformal. After presenting some examples, we classify the conformal vector fields on two famous statistical manifolds. Considering three statistical structures on the tangent bundle of a statistical manifold, we study the conditions under which the complete and horizontal lifts of a vector field can be conformal on these structures.
The isometry groups of Lorentzian three-dimensional unimodular simply connected Lie groups. Mohamed Boucetta and Abdelmounaim Chakkar
We determine the isometry groups of all three-dimensional, connected, simply connected and unimodular Lie groups endowed with a left-invariant Lorentzian metric.
On the differential and Volterra-type integral operators on Fock-type spaces. Tesfa Mengestie
The differential operator fails to admit some basic structures including continuity when it acts on the classical Fock spaces or weighted Fock spaces, where the weight functions grow faster than the classical Gaussian weight function. The same conclusion also holds in some weighted Fock spaces including the Fock–Sobolev spaces, where the weight functions grow more slowly than the Gaussian function. We consider modulating the classical weight function and identify Fock-type spaces where the operator admits the basic structures. We also describe some properties of Volterra-type integral operators on these spaces using the notions of order and type of entire functions. The modulation operation supplies richer structures for both the differential and integral operators in contrast to the classical setting.
A note on the density of the partial regularity result in the class of viscosity solutions. Disson dos Prazeres, Edgard A. Pimentel, and Giane C. Rampasso
We establish the density of the partial regularity result in the class of continuous viscosity solutions. Given a fully nonlinear equation, we prove the existence of a sequence entitled to the partial regularity result, approximating its solutions. Distinct conditions on the operator driving the equation lead to density in different topologies. Our findings include applications to nonhomogeneous problems, with variable-coefficient models.
Monster graphs are determined by their Laplacian spectra. Ali Zeydi Abdian, Ali Reza Ashrafi, Lowell W. Beineke, Mohammad Reza Oboudi, and Gholam Hossein Fath-Tabar
A graph $G$ is determined by its Laplacian spectrum (DLS) if every graph with the same Laplacian spectrum is isomorphic to $G$. A multi-fan graph is a graph of the form $(P_{n_1}\cup P_{n_2}\cup \cdots \cup P_{n_k})\bigtriangledown K_1$, where $K_1$ denotes the complete graph of size 1, $P_{n_1}\cup P_{n_2}\cup \cdots \cup P_{n_k}$ is the disjoint union of paths $P_{n_i}$, $n_i\geq 1$ and $1 \leq i \leq k$; and a starlike tree is a tree with exactly one vertex of degree greater than 2. If a multi-fan graph and a starlike tree are joined by identifying their vertices of degree more than 2, then the resulting graph is called a monster graph. In some earlier works, it was shown that all multi-fan and path-friendship graphs are DLS. The aim of this paper is to generalize these facts by proving that all monster graphs are DLS.
Sharp bounds for fractional type operators with $L^{\alpha,s}$-Hörmander conditions. Gonzalo H. Ibañez-Firnkorn, María Silvina Riveros, and Raúl E. Vidal
We provide the sharp bound for a fractional type operator given by a kernel satisfying the $L^{\alpha,s}$-Hörmander condition and certain fractional size condition, $0 < \alpha < n$ and $1 < s\leq \infty$. In order to prove this result we use a new appropriate sparse domination. Examples of these operators include the fractional rough operators. For the case $s=\infty$ we recover the sharp bound of the fractional integral, $I_{\alpha}$, proved by Lacey et al. [J. Functional Anal. 259 (2010), no. 5, 1073–1097].
The isolation of the first eigenvalue for a Dirichlet eigenvalue problem involving the Finsler $p$-Laplacian and a nonlocal term. Andrei Grecu
We analyse the isolation of the first eigenvalue for an eigenvalue problem involving the Finsler $p$-Laplace operator and a nonlocal term on a bounded domain subject to the homogeneous Dirichlet boundary condition.
The algebraic sets of vectors generating planar normal sections of isoparametric hypersurfaces of FKM type. Cristián U. Sánchez
We present a proof of the connectedness of the algebraic set of vectors generating planar normal sections for all isoparametric hypersurfaces, with positive multiplicities $m_{1}$ and $m_{2}$ of FKM type.
Orlicz version of mixed moment tensors. Chang-Jian Zhao
Our main aim is to generalize the moment tensors $\mathbf{\Psi}_{r}(K)$ to the Orlicz space. Under the framework of the Orlicz–Brunn–Minkowski theory, we introduce a new affine geometric quantity $\mathbf{\Psi}_{\psi,r}(K,L)$, and call it Orlicz mixed moment tensors of convex bodies $K$ and $L$. The fundamental notions and properties of the moment tensors as well as related Minkowski and Brunn–Minkowski inequalities are then extended to the Orlicz setting. Diverse inequalities for certain new $L_p$-mixed moment tensors $\mathbf{\Psi}_{p,r}(K,L)$ are also derived. The new Orlicz inequalities in special cases yield the Orlicz–Minkowski and the Orlicz–Brunn–Minkowski inequalities, respectively.
Affinity kernels on measure spaces and maximal operators. Hugo Aimar, Ivana Gómez, and Luis Nowak
In this note we consider maximal operators defined in terms of families of kernels and families of their level sets. We prove a general estimate that extends some classical Euclidean cases and, under some mild transitivity property, we show their basic boundedness properties on Lebesgue spaces. The motivation of these problems has its roots in the analysis associated to affinity kernels on large data sets.
Bilinear differential operators and $\mathfrak{osp}(1|2)$-relative cohomology on $\mathbb{R}^{1|1}$. Abderraouf Ghallabi and Meher Abdaoui
We consider the $1|1$-dimensional real superspace $\mathbb{R}^{1|1}$ endowed with its standard contact structure defined by the 1-form $\alpha$. The conformal Lie superalgebra $\mathcal{K}(1)$ acts on $\mathbb{R}^{1|1}$ as the Lie superalgebra of contact vector fields; it contains the Möbius superalgebra $\mathfrak{osp}(1|2)$. We classify $\mathfrak{osp}(1|2)$-invariant superskew-symmetric binary differential operators from $\mathcal{K}(1)\wedge\mathcal{K}(1)$ to $\mathfrak{D}_{\lambda,\mu;\nu}$ vanishing on $\mathfrak{osp}(1|2)$, where $\mathfrak{D}_{\lambda,\mu;\nu}$ is the superspace of bilinear differential operators between the superspaces of weighted densities. This result allows us to compute the second differential $\mathfrak{osp}(1|2)$-relative cohomology of $\mathcal{K}(1)$ with coefficients in $\mathfrak{D}_{\lambda,\mu;\nu}$.
On the common fixed points of double sequences of two-parameter nonexpansive semigroups in strictly convex Banach spaces. Rasoul Abazari and Asadollah Niknam
We obtain some results about common fixed points of strongly continuous two-parameter semigroups of nonexpansive mappings in strictly convex Banach spaces. Also, by using them we characterize common fixed points of double sequences of such semigroups.