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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 co-recursive orthogonal polynomials with respect to a quasi-definite moment functional $L_{k}$. We find an integral representation for $L_{k}$ and compute explicit expressions for all of its moments. 1–30 Time-frequency 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 Borel-stable 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 zero-divisor 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 zero-divisors of $R$. The weakly zero-divisor 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 zero-divisor 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 zero-divisor graphs are star. We also give conditions under which weakly zero-divisor and zero-divisor 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 2-algebras of odd toral rank and a contragredient Lie 2-algebra 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 matrix-valued functions instead of scalar-valued 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 square-integrable 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 Hom-Jordan algebras. Chenrui Yao, Yao Ma, and Liangyun Chen We study the structure of simple multiplicative Hom-Jordan algebras. We discuss equivalent conditions for multiplicative Hom-Jordan algebras to be solvable, simple, and semi-simple. Moreover, we give a theorem on the classification of simple multiplicative Hom-Jordan algebras and obtain some propositions about bimodules of multiplicative Hom-Jordan 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 Papari-Zarei Let $n$ be a non-negative 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 semi-local 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 1-type 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 3-space. We introduce the notion of generalized 1-type 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 non-zero smooth functions. First of all, we show that the Gauss map of any surface of revolution with unit speed profile curve in Euclidean 3-space is of generalized 1-type. At the same time, the canal surfaces with generalized 1-type Gauss map are discussed. 199–211 Selberg zeta-function associated to compact Riemann surface is prime. Ramūnas Garunkštis Let $Z(s)$ be the Selberg zeta-function 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 $y^p-y = xR(x) + \beta x$, $p$ 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 four-dimensional homogeneous manifolds. Milad Bastami, Ali Haji-Badali, and Amirhesam Zaeim Recurrent curvature properties are considered on four-dimensional pseudo-Riemannian homogeneous manifolds with non-trivial 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