Revista de la
Unión Matemática Argentina

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Articles are published here before the issue is completed.

Vol. 68, no. 2 (2025)

On the second $\mathfrak{osp}(1|2)$-relative cohomology of the Lie superalgebra of contact vector fields on $\mathcal{C}^{1|1}$. Abderraouf Ghallabi, Nizar Ben Fraj, and Salem Faidi
Let $\mathcal{K}(1)$ be the Lie superalgebra of contact vector fields on the $(1,1)$-dimensional complex superspace; it contains the Möbius superalgebra $\mathfrak{osp}(1|2)$. We classify $\mathfrak{osp}(1|2)$-invariant superanti-symmetric binary differential operators from $\mathcal{K}(1)\wedge\mathcal{K}(1)$ to $\mathfrak{D}_{\lambda,\mu}$ vanishing on $\mathfrak{osp}(1|2)$, where $\mathfrak{D}_{\lambda,\mu}$ is the superspace of linear differential operators acting on 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 $\mathfrak{D}_{\lambda,\mu}$.
349–367
Clones from comonoids. Ulrich Krähmer and Myriam Mahaman
We revisit the fact that the cocommutative comonoids in a symmetric monoidal category form the best possible approximation by a cartesian category, now considering the case where the original category is only braided monoidal. This leads to the question of when the endomorphism operad of a comonoid is a clone (a Lawvere theory). By giving an explicit example, we prove that this does not imply that the comonoid is cocommutative.
369–394
Frobenius property for fusion categories of dimension 120. Li Dai
We prove that fusion categories of Frobenius–Perron dimensions 120 are of Frobenius type. Combining this with known results in the literature, we get that all weakly integral fusion categories of Frobenius–Perron dimension less than 126 are of Frobenius type.
395–403
Linear functionals and $\Delta$-coherent pairs of the second kind. Diego Dominici and Francisco Marcellán
We classify all the $\Delta$-coherent pairs of measures of the second kind on the real line. We obtain five cases, corresponding to all the families of discrete semiclassical orthogonal polynomials of class $s\leq1$.
405–422
Finite groups in which some maximal subgroups are MNP-groups. Pengfei Guo and Huaguo Shi
A finite group $G$ is called an MNP-group if all maximal subgroups of the Sylow subgroups of $G$ are normal in $G$. The aim of this paper is to give a necessary and sufficient condition for a group to be an MNP-group, characterize the structure of finite groups whose maximal subgroups (respectively, maximal subgroups of even order) are all MNP-groups, and determine finite non-abelian simple groups whose second maximal subgroups (respectively, maximal subgroups of even order) are all MNP-groups.
423–435
A canonical distribution on isoparametric submanifolds III. Cristián U. Sánchez
The present paper is devoted to showing that on every compact, connected homogeneous isoparametric submanifold $M=G/K$ of codimension $h\geq2$ in a Euclidean space, there exist canonical distributions which are generated by the compact symmetric spaces associated to $M$ (i.e., those corresponding to the group $G$). The central objective is to show that all these distributions are bracket generating of step 2. To that end, formulae that complement those in the first article of this series (Rev. Un. Mat. Argentina 61, no. 1 (2020), 113–130) are obtained.
437–458
New harmonic-measure distribution functions of some simply connected planar regions in the complex plane. Arunmaran Mahenthiram
Consider a Brownian particle released from a fixed point $z_0$ in a region $\Omega$. The harmonic-measure distribution function, or $h$-function, $h(r)$, expresses the probability that the Brownian particle first hits the boundary $\partial\Omega$ of the region $\Omega$ within distance $r$ of $z_0$. In this paper, we compute the $h$-function of several new planar simply connected two-dimensional regions by using two different methods, both involving conformal maps. We also explain the asymptotic behaviour at certain values of $r$ where two different regimes meet. Moreover, for some regions, we examine how the behaviour of $h(r)$ changes when part of the boundary changes.
459–483
On the moduli space of left-invariant metrics on the cotangent bundle of the Heisenberg group. Tijana Šukilović, Srdjan Vukmirović, and Neda Bokan
The focus of the paper is on the study of the moduli space of left-invariant pseudo- Riemannian metrics on the cotangent bundle of the Heisenberg group. We use algebraic approach to obtain orbits of the automorphism group acting in a natural way on the space of left invariant metrics. However, geometric tools such as the classification of hyperbolic plane conics are often needed. For the metrics obtained by the classification, we study geometric properties: curvature, Ricci tensor, sectional curvature, holonomy, and parallel vector fields. The classification of algebraic Ricci solitons is also presented, as well as the classification of pseudo-Kähler and pp-wave metrics. We obtain description of parallel symmetric tensors for each metric and show that they are derived from parallel vector fields. Finally, we study the totally geodesic subalgebras and show that for each subalgebra of the observed algebra there is a metric which makes it totally geodesic.
485–518
Regular automorphisms and Calogero–Moser families. Cédric Bonnafé
We study the subvariety of fixed points of an automorphism of a Calogero–Moser space induced by a regular element of finite order of the normalizer of the associated complex reflection group $W$. We determine some of (and conjecturally all) the ${\mathbb{C}}^\times$-fixed points of its unique irreducible component of maximal dimension in terms of the character table of $W$. This is inspired by the mysterious relations between the geometry of Calogero–Moser spaces and unipotent representations of finite reductive groups, which is the theme of another paper [Pure Appl. Math. Q. 21 no. 1 (2025), 131–200].
519–533
Graded almost valuation rings. Fatima Zahra Guissi, Najib Mahdou, Ünsal Tekir, and Suat Koç
Let $R=\bigoplus_{\alpha\in\Gamma}R_{\alpha}$ be a commutative ring graded by an arbitrary torsionless monoid $\Gamma$. We say that $R$ is a graded almost valuation ring (gr AV- ring) if for every two homogeneous elements $a,b$ of $R$, there exists a positive integer $n$ such that either $a^{n}$ divides $b^{n}$ (in $R$) or $b^{n}$ divides $a^{n}$. In this paper, we introduce and study the graded version of the almost valuation ring which is a generalization of gr-AVD to the context of arbitrary $\Gamma$-graded rings (with zero- divisors). Next, we study the possible transfer of this property to the graded trivial ring extension $A\ltimes E$. Our aim is to provide examples of new classes of $\Gamma$-graded rings satisfying the above mentioned property.
535–553
A generalized Bernoulli differential equation. Hector Carmenate, Paul Bosch, Juan E. Nápoles, and José M. Sigarreta
We study a generalized form of the Bernoulli differential equation, employing a generalized conformable derivative. We first establish a generalized variant of Gronwall's inequality, which is essential for assessing the stability of generalized differential equation systems, and offer insights into the qualitative behavior of the trivial solution of the proposed equation. We then present and prove the main results concerning the solution of the generalized Bernoulli differential equation, complemented by illustrative examples that highlight the advantages of this generalized derivative approach. Furthermore, we introduce a finite difference method as an alternative technique to approximate the solution of the generalized Bernoulli equation and demonstrate its validity through practical examples.
555–575
The pointillist principle for variation operators and jump functions. Kevin Hughes
I extend the pointillist principles of Moon and Carrillo–de Guzmán to variational operators and jump functions.
577–588
A high-accuracy compact finite difference scheme for time-fractional diffusion equations. Xindong Zhang, Hanxiao Wang, Ziyang Luo, and Leilei Wei
We propose a compact finite difference (CFD) scheme for the solution of time-fractional diffusion equations (TFDE) with the Caputo–Fabrizio derivative. The Caputo–Fabrizio derivative is discussed in the time direction and is discretized by a special discrete scheme. The compact difference operator is introduced in the space direction. We prove the unconditional stability and convergence of the proposed scheme. We show that the convergence order is $O(\tau^3+h^4)$, where $\tau$ and $h$ are the temporal stepsize and spatial stepsize, respectively. Our main purpose is to show that the Caputo–Fabrizio derivative without singular term can improve the accuracy of the discrete scheme. Numerical examples demonstrate the efficiency of the proposed method, and the numerical results agree well with the theoretical predictions.
589–609
Hamiltonicity of rectangular grid graphs (meshes) with an L-shaped hole. Movahedeh Rouhani-Marchoobeh and Fatemeh Keshavarz-Kohjerdi
Finding the Hamiltonian cycles in graphs is a well-known problem. Although the Hamiltonicity of grid graphs has been studied in the literature, there are few results on Hamiltonicity of grid graphs with holes. In this paper, we study the Hamiltonicity of rectangular grid graphs (meshes) with an L-shaped hole, and give a linear-time algorithm. The holes in meshes correspond to the faulty nodes.
611–625
Conditional non-lattice integration, pricing, and superhedging. Christian Bender, Sebastian E. Ferrando, and Alfredo L. Gonzalez
Motivated by financial considerations, we develop a non-classical integration theory that is not necessarily associated with a measure. The base space consists of stock price trajectories and embodies a natural no-arbitrage condition. Conditional integrals are introduced, representing the investment required to hedge an option payoff when entering the market at any later time. Here, the investment may depend on the stock price history, and hedging takes place almost everywhere and as a limit over an increasing number of portfolios. In our setting, the space of elementary integrands fails to satisfy the lattice property and the notion of null sets is financially motivated and not measure- theoretic. Therefore, option prices arise from conditional non-lattice integrals rather than expectations, with no need to impose measurability assumptions.
627–676
Depth and Stanley depth of powers of the path ideal of a cycle graph. Silviu Bălănescu and Mircea Cimpoeaş
Let $J_{n,m}:=(x_1x_2\cdots x_m, x_2x_3\cdots x_{m+1}, \ldots, x_{n-m+1}\cdots x_n, x_{n-m+2}\cdots x_nx_1, \ldots, x_nx_1\cdots x_{m-1})$ be the $m$-path ideal of the cycle graph of length $n$ in the ring $S=K[x_1,\ldots,x_n]$. Let $d=\gcd(n,m)$. We prove that $\operatorname{depth}(S/J_{n,m}^t)\leq d-1$ for all $t\geq n-1$. We show that $\operatorname{sdepth}(S/J_{n,n-1}^t)=\operatorname{depth}(S/J_{n,n-1}^t)=\max\{n-t-1,0\}$ for all $t\geq 1$. Also, we give some bounds for $\operatorname{depth}(S/J_{n,m}^t)$ and $\operatorname{sdepth}(S/J_{n,m}^t)$, where $t\geq 1$.
677–690
On a non-standard characterization of the $A_p$ condition. Andrei K. Lerner
The classical Muckenhoupt $A_p$ condition is necessary and sufficient for the boundedness of the maximal operator $M$ on $L^p(w)$ spaces. In this paper we obtain another characterization of the $A_p$ condition. As a result, we show that some strong versions of the weighted $L^p(w)$ Coifman–Fefferman and Fefferman–Stein inequalities hold if and only if $w\in A_p$. We also give new examples of Banach function spaces $X$ such that $M$ is bounded on $X$ but not bounded on the associate space $X'$.
691–701
The conjecture on distance-balancedness of generalized Petersen graphs holds when internal edges have jumps 3 or 4. Gang Ma, Jianfeng Wang, and Sandi Klavžar
A connected graph $G$ with $\mathrm{diam}(G) \ge \ell$ is $\ell$-distance-balanced if $|W_{xy}|=|W_{yx}|$ for every $x,y\in V(G)$ with $d_{G}(x,y)=\ell$, where $W_{xy}$ is the set of vertices of $G$ that are closer to $x$ than to $y$. Miklavič and Šparl [Discrete Appl. Math. 244 (2018), 143–154] conjectured that if $n>n_k$, where $n_k=11$ if $k=2$, $n_k=(k+1)^2$ if $k$ is odd, and $n_k=k(k+2)$ if $k\ge 4$ is even, then the generalized Petersen graph $\mathrm{GP}(n,k)$ is not $\ell$-distance-balanced for any $1\le \ell < \mathrm{diam}(\mathrm{GP}(n,k))$. In the seminal paper, the conjecture was verified for $k=2$. In this paper we prove that the conjecture holds for $k=3$ and for $k=4$.
703–733
On $L_p$ Ky Fan determinant inequalities. Bingxiu Lyu and Danni Xu
We establish an extension of Ky Fan's determinant inequality when the usual matrix addition is replaced by the power mean of positive definite matrices. We further explore variants of this newly derived $L_p$ Ky Fan inequality, extending a determinant difference inequality formulated by Yuan and Leng [J. Aust. Math. Soc. 83 no. 1 (2007)].
735–744
The Newman algorithm for constructing polynomials with restricted coefficients and many real roots. Markus Jacob and Fedor Nazarov
Under certain natural sufficient conditions on the sequence of uniformly bounded closed sets $E_k\subset\mathbb{R}$ of admissible coefficients, we construct a polynomial $P_n(x)=1+\sum_{k=1}^n\varepsilon_k x^k$, $\varepsilon_k\in E_k$, with at least $c\sqrt n$ distinct roots in $[0,1]$, which matches the classical upper bound up to the value of the constant $c>0$. Our sufficient conditions cover the Littlewood ($E_k=\{-1,1\}$) and Newman ($E_k=\{0,(-1)^k\}$) polynomials and are also necessary for the existence of such polynomials with arbitrarily many roots in the case when the sequence $E_k$ is periodic.
745–759
Superpower graphs of finite abelian groups. Ajay Kumar, Lavanya Selvaganesh, and T. Tamizh Chelvam
For a finite group $G$, the superpower graph $S(G)$ is a simple undirected graph with vertex set $G$, where two distinct vertices are adjacent if and only if the order of one divides that of the other. The aim of this paper is to provide tight bounds for the vertex connectivity of $S(G)$, together with some structural properties such as maximal domination sets, Hamiltonicity, and its variations for superpower graphs of finite abelian groups. The paper concludes with some open problems.
761–773
Recurrence for weighted pseudo-shift operators. Mohamed Amouch and Fatima-ezzahra Sadek
We provide a characterization of multiply recurrent operators that act on a Fréchet space. As an application, we extend the weighted shift results established by Costakis and Parissis (2012). We achieve this by characterizing topologically multiply recurrent pseudo-shifts acting on an $F$-sequence space indexed by an arbitrary countable infinite set. This characterization is in terms of the weights, the OP-basis and the shift mapping. Additionally, we establish that the recurrence and the hypercyclicity of pseudo-shifts are equivalent.
775–786