Issue 2, Volume 48, June 2023

2023

1. Discrete Fourier-Jacobi Transform and Generalized Lipschitz Classes El Mehdi Loualid , Abdelghani Elgargati , Radouan Daher PDF

   ▸ Abstract

   In this paper, we use the methods of Fourier-Jacobi harmonic analysis to generalize Boas-type results. We give necessary and sufficient conditions in terms of the Fourier-Jacobi coefficients of a function $f$ in order to ensure that it belongs either to one of the generalized Lipschitz classes ${H}_{\alpha }^{m}$ and ${h}_{\alpha }^{m}$ for $\alpha >  0$.

   259-269
2. Measured Foliations and Hilbert 12th Problem Igor V. Nikolaev PDF

   ▸ Abstract

   Yu. I. Manin conjectured that the maximal abelian extensions of the real quadratic number fields are generated by the pseudo-lattices with real multiplication. We prove this conjecture using theory of measured foliations on the modular curves.

   271-281
3. On Monogenity of Certain Pure Number Fields Defined by $x^{60} − m$ Lhoussain El Fadil , Hanan Choulli , Omar Kchit PDF

   ▸ Abstract

   Let $K$ be a pure number field generated by a complex root of a monic irreducible polynomial $F(x)=x^{60}-m\in \mathbb{Z}[x]$, with $m\neq \pm1$ a square free integer. In this paper, we study the {monogenity} of $K$. We prove that if $m\not\equiv 1~(\mathrm{mod}~{4})$, $m\not\equiv \pm 1 ~(\mathrm{mod}~{9}) $ and $\overline{m}\not\in\{\pm 1,\pm 7\} ~({\rm mod}~{25})$, then $K$ is monogenic. But if $m\equiv 1~({\rm mod}~{4})$, $m\equiv \pm1 ~({\rm mod}~{9})$, or $m\equiv \pm 1~({\rm mod}~{25})$, then $K$ is not monogenic. Our results are illustrated by examples.

   283-293
4. Finite Decomposition of Herz-Type Hardy Spaces for the Dunkl Operator Mehdi Lachiheb PDF

   ▸ Abstract

   The corresponding Herz-type Hardy spaces to new weighted Herz spaces $HK^{\beta ,p}_{\alpha ,q}$ associated with the Dunkl operator on $\mathbb R$ have been characterized by atomic decompositions. Later a new characterization of $HK^{\beta ,p}_{\alpha ,q}$ on the real line is introduced. This helped us in the work to characterize that the norms of the Herz-type Hardy spaces for the Dunkl Operator can be achieved by finite central atomic decomposition in some dense subspaces of them. Secondly, as an application we prove that a sublinear operator satisfying many conditions can be uniquely extended to a bounded operator in the Herz-type Hardy spaces for the Dunkl Operator.

   295-306
5. The Existence of Balanced Neighborly Polynomials Nguyen Thi Thanh Tam PDF

   ▸ Abstract

   Inspired by the definition of balanced neighborly spheres, we define balanced neighborly polynomials and study the existence of these polynomials. The goal of this article is to construct balanced neighborly polynomials of type $(k,k,k,k)$ over any field $K$ for all $k\not = 2$, and show that a balanced neighborly polynomial of type $(2,2,2,2)$ exists if and only if $\mathrm{char}(K)\not =  2$. Besides, we also discuss a relation between balanced neighborly polynomials and balanced neighborly simplicial spheres.

   307-319
6. Necessary Optimality Conditions for Second-Order Local Strict Efficiency for Constrained Nonsmooth Vector Equilibrium Problems Tran Van Su , Dinh Dieu Hang PDF

   ▸ Abstract

   This paper is concerned with primal and dual second-order optimality conditions for the second-order strict efficiency of nonsmooth vector equilibrium problem with set, cone and equality conditions. First, we propose some second-order constraint qualifications via the second-order tangent sets. Second, we establish necessary optimality conditions of order two in terms of second-order contingent derivatives and second-order Shi sets for a second-order strict local Pareto minima to such problem under suitable assumptions on the second-order constraint qualifications. An application of the result for the twice Fréchet differentiable functions for the second-order local strict efficiency of that problem is also presented. Some illustrative examples are also provided for our findings.

   321-341
7. Some Properties of $n$-semidualizing Modules Tony Se PDF

   ▸ Abstract

   Let $R$ be a commutative noetherian ring. The $n$-semidualizing modules of $R$ are generalizations of its semidualizing modules. We will prove some basic properties of $n$-semidualizing modules. Our main result and example shows that the divisor class group of a Gorenstein determinantal ring over a field is the set of isomorphism classes of its 1-semidualizing modules. Finally, we pose some questions about $n$-semidualizing modules.

   343-358
8. Hilbert Genus Fields of Some Number Fields with High Degrees Mohamed Mahmoud Chems-Eddin , Moulay Ahmed Hajjami , Mohammed Taous PDF

   ▸ Abstract

   The aim of this paper is to give some properties of Hilbert genus fields and construct the Hilbert genus fields of the fields $L_{m,d}:=\mathbb {Q}(\zeta _{2^{m}},\sqrt {d})$, where $m \ge 3$ is a positive integer and $d$ is a square-free integer whose prime divisors are congruent to $\pm 3$ (mod 8) or 9 (mod 16).

   359-369
9. Continuous Solutions for Degenerate Complex Hessian Equation Hichame Amal , Saïd Asserda , Manar Bouhssina PDF

   ▸ Abstract

   Let $(X,\omega)$ be an $n$-dimensional compact Kähler manifold and fix an integer $m$ such that $1\leq m\leq n$. Let $\mu$ be a finite Borel measure on $X$ satisfying the conditions $\mathcal{H}_m(\delta, A,\omega)$. We study degenerate complex Hessian equations of the form $(\omega+ dd^{c}\varphi)^{m}\wedge\omega^{n-m}=F(\varphi,.)d\mu$. Under some natural conditions on $F$, we prove that if $0<\delta<\frac{m}{n-m}$, then this equation has a unique continuous solution.

   371-386