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.

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 $L^{p}_{\alpha}(\mathbb{K})$, $1 \leq p \leq \infty$.

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.

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.

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.

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.

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].

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 $G(F_{4, a})$.

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.

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.

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.

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.

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.

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.

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 $\mathcal{L} = \langle x^{p^\ell}, x^{p^{3\ell}} \rangle$.

Recurrent curvature properties are considered on four-dimensional pseudo-Riemannian homogeneous manifolds with non-trivial isotropy, and also on some geometric manifolds.

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 $F(n) = \#\{(a,m_1,m_2,m_3) : \frac{a}{n}=\frac{1}{m_1}+\frac{1}{m_2}+\frac{1}{m_3}\}$.

Motivated by the $F$-contraction introduced by Wardowski [Fixed Point Theory Appl. 2012, 2012:94], we introduce three types of multidimensional Ciric-type rational $F$-contractions for multivalued mappings in dislocated metric spaces. Using these contractions, we establish fixed points of $N$-order for multivalued mappings. Our result generalizes the main result obtained by Rasham et al. [J. Fixed Point Theory Appl. 20 (2018), no. 1, Paper no. 45].

This paper is devoted to presenting ordinal sums of MTL-chains as a particular case of split short exact sequences of finite chains in the category of semihoops. This module theoretical approach will allows us to prove, in a very elementary way, that every finite locally unital MTL-chain can be decomposed as an ordinal sum of archimedean MTL-chains. Furthermore, we propose the study of MTL-chain extensions and we show that ordinal sums of locally unital MTL-chains are a particular case of these.

We propose a modified Halpern-type algorithm involving a Lipschitz hemicontractive non-self mapping and the resolvent of a convex function in a Hadamard space. We obtain a strong convergence of the proposed algorithm to a minimizer of a convex function which is also a fixed point of a Lipschitz hemicontractive non-self mapping. Furthermore, we give a numerical example to illustrate and support our method. Our proposed method improves and extends some recent works in the literature.

Locally, a screen integrable globally null manifold $M$ splits through a Riemannian leaf $M'$ of its screen distribution and a null curve $\mathcal{C}$ tangent to its radical distribution. The leaf $M'$ carries a lot of geometric information about $M$ and, in fact, forms a basis for the study of expanding and non-expanding horizons in black hole theory. In the present paper, we introduce a degenerate Ricci-type flow in $M'$ via the intrinsic Ricci tensor of $M$. Several new gradient estimates regarding the flow are proved.

Let $F$ be a product preserving gauge bundle functor on double vector bundles. We introduce the complete lifting $\mathcal{F}\varphi:FK\to \wedge^p T^*FM\otimes TFK$ of a double-linear semi-basic tangent valued $p$-form $\varphi:K\to \wedge^p T^*M\otimes TK$ on a double vector bundle $K$ with base $M$. We prove that this complete lifting preserves the Frolicher–Nijenhuis bracket. We apply the results obtained to double-linear connections.

This paper is devoted to investigating the notions of simple, local and subdirectly irreducible state residuated lattices and some of their related properties. The filters generated by a subset in state residuated lattices are characterized and it is shown that the lattice of filters of a state residuated lattice forms a complete Heyting algebra. Maximal, prime and minimal prime filters of a state residuated lattice are investigated and it is shown that any filter of a state residuated lattice contains a minimal prime filter. Finally, the relevant notions are discussed and characterized.

We first give the notion of Reeb invariant Ricci tensor for real hypersurfaces $M$ in the complex quadric ${Q^m}^*=SO^0_{2,m}/SO_2 SO_m$, which is defined by $\mathcal{L}_{\xi}\operatorname{Ric}=0$, where $\operatorname{Ric}$ denotes the Ricci tensor of $M$ in ${Q^m}^*$, and $\mathcal{L}_{\xi}$ the Lie derivative along the direction of the Reeb vector field $\xi=-JN$. Next we give a complete classification of real hypersurfaces in the complex hyperbolic quadric ${Q^m}^*=SO^0_{2,m}/SO_2 SO_m$ with Reeb invariant Ricci tensor.

We use the Henstock–Kurzweil integral and interpolation theory to extend the Fourier cosine transform operator, broadening some classical properties such as the Riemann–Lebesgue lemma. Furthermore, we show that a qualitative difference between the cosine and sine transform is preserved on differentiable functions.

Let $R$ be a local ring and suppose that there exists $a\in F^*$ such that $a^6\neq 1$; also let $T: M_n(R) \to M_m(R)$ be a linear map preserving Drazin inverses. Then we prove that $T=0$ or $n=m$ and $T$ preserves idempotents. We thereby determine the form of linear maps from $M_n(R)$ to $M_m(R)$ preserving Drazin inverses of matrices.

We present $*$-primary submodules, a generalization of the concept of primary submodules of an $R$-module. We show that every primary submodule of a Noetherian $R$-module is $*$-primary. Among other things, we show that over a commutative domain $R$, every torsion free $R$-module is $*$-primary. Furthermore, we show that in a cyclic $R$-module, primary and $*$-primary coincide. Moreover, we give a characterization of $*$-primary submodules for some finitely generated free $R$-modules.

We characterize the local superderivations on Cartan type Lie superalgebras over the complex field $\mathbb{C}$. Furthermore, we prove that every local superderivation on Cartan type Lie superalgebras is a superderivation. As an application, using the results on local superderivations we characterize the linear $2$-local superderivations on Cartan type Lie superalgebras. We prove that every linear $2$-local superderivation on Cartan type Lie superalgebras is a superderivation.

We introduce a definition of a generalized conformable derivative of order $\alpha \gt 0$ (where this parameter does not need to be integer), with which we overcome some deficiencies of known local derivatives, conformable or not. This definition allows us to compute fractional derivatives of functions defined on any open set on the real line (and not just on the positive half-line). Moreover, we extend some classical results to the context of fractional derivatives. Also, we obtain results for the case $\alpha \gt 1$.

In this paper, we introduce a new bounded distribution by using trigonometric functions, named the cosine-sine distribution. A comprehensive study of its statistical properties is presented along with an application to a unit-interval data set, namely firms risk management cost-effectiveness data. The proposed distribution has increasing, bathtub and v hazard rate shapes. Further, we show that the distribution can be viewed as a truncated exponential sine distribution.

In this work we study an indefinite abstract smoothing problem. After establishing necessary and sufficient conditions for the existence of solutions, the set of admissible parameters is discussed in detail. Then, the relationship of this problem with a linearly constrained interpolation problem is analyzed.

The present paper continues our previous work [Rev. Un. Mat. Argentina 61 (2020), no. 1, 113–130], which was devoted to showing 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. In that work this fact was established for the case when the system of restricted roots is reduced. Here we complete the proof of the main result for the case in which the system of restricted roots is $(BC)_{q}$, i.e., non-reduced.