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This paper studies the transfer of pseudo-almost valuation property (PAVR property for short) to various context of commutative ring extensions such as power series ring, trivial ring extension and amalgamation. Our work is motivated by an attempt to generate new original classes of rings satisfying this property. The obtained results are backed with new and illustrative examples arising as trivial ring extensions, amalgamations and pullback constructions.

In this paper, we study the almost Bézout property in different commutative ring extensions, namely, in bi-amalgamated algebras and pairs of rings. In Section 2, we deal with almost Bézout domains issued from bi-amalgamations. Our results capitalize well known results on amalgamations and pullbacks as well as generate new original class of rings satisfying this property. Section 3 investigates pairs of rings where all intermediate rings are almost Bézout domains. As an application of our results, we characterize pairs of rings $(R,T)$, where $R$ arises from a $(T,M,D)$ construction to be an almost Bézout domain.

In this paper, we discuss new results related to the generalized discrete q-Hermite II polynomials h ˜ n , α ( x ; q ) , introduced by Mezlini et al. in 2014. Our aim is to give a continuous orthogonality relation, a q-integral representation and an evaluation at unity of the Poisson kernel, for these polynomials. Furthermore, we introduce q-Schrodinger operators and we give the raising and lowering operator algebra corresponding to these polynomials. Our results generate a new explicit realization of the quantum algebra su q ( 1 , 1 ) , using the generators associated with a q-deformed generalized para-Bose oscillator.

This paper examines the transfer of the almost valuation property to various constructions of ring extensions such as bi-amalgamations and pairs of rings. Namely, Sec. 2 studies the transfer of this property to bi-amalgamation rings. Our results cover previous known results on duplications and amalgamations, and provide the construction of various new and original examples satisfying this property. Section 3 investigates pairs of integral domains where all intermediate rings are almost valuation rings. As a consequence of our results, we provide necessary and sufficient conditions for a pair (R, T), where R arises from a (T, M, D) construction, to be an almost valuation pair. Furthermore, we study the class of maximal non-almost valuation subrings of their quotient field.

Let Γ ⊂ PSL (2, ℝ) be a discrete and finite covolume subgroup. We suppose that the modular curve [Formula: see text] is "embedded" in a Hilbert modular surface [Formula: see text], where ΓK is the Hilbert modular group associated to a real quadratic field K. We define a sequence of restrictions (ρn)n∈ℕ of Hilbert modular forms for ΓK to modular forms for Γ. We denote by Mk1, k2(ΓK) the space of Hilbert modular forms of weight (k1, k2) for ΓK. We prove that ∑n∈ℕ ∑k1, k2∈ℕ ρn(Mk1, k2(ΓK)) is a subalgebra closed under Rankin–Cohen brackets of the algebra ⊕k∈ℕ Mk(Γ) of modular forms for Γ.

Resume Soit Γ ⊂ PSL ( 2 , R ) un sous-groupe discret de covolume fini. On suppose que la courbe modulaire H / Γ se « plonge » dans une surface modulaire de Hilbert H 2 / Γ K , ou Γ K est le groupe modulaire de Hilbert associe a un corps quadratique reel K . On definit une suite de restrictions ( ρ n ) n ∈ N des formes modulaires de Hilbert pour Γ K aux formes modulaires pour Γ . Notons M k 1 , k 2 ( Γ K ) l'espace des formes modulaires de Hilbert de poids ( k 1 , k 2 ) pour Γ K . On demontre que ∑ n ∈ N ∑ k 1 , k 2 ∈ N ρ n ( M k 1 , k 2 ( Γ K ) ) est un sous-espace stable par les crochets de Rankin–Cohen de l'espace ⊕ k ∈ N M k ( Γ ) des formes modulaires pour Γ . Pour citer cet article : N. Ouled Azaiez, C. R. Acad. Sci. Paris, Ser. I 344 (2007).

Resume On decrit la structure additive de l'anneau gradue M ˜ ∗ ( Γ ) des formes quasi-modulaires sur un groupe discret et co-compact Γ ⊂ PSL ( 2 , R ) . On montre que cet anneau n'est pas de type fini. On calcule le nombre de generateurs nouveaux en chaque poids k (pair). Le nombre en question est fixe pour k assez grand, et est egal a dim C I / ( I ∩ I ˜ 2 ) ou I et I ˜ designent les ideaux des formes modulaires, respectivement quasi-modulaires, sur Γ en poids strictement positif. Pour citer cet article : N. Ouled Azaiez, C. R. Acad. Sci. Paris, Ser. I 343 (2006).

The aim of this work is to prove new results on a class of digital functions with special emphasis on shifted primes as arguments. Our method lies on the estimate of exponential sums of the form $\sum_{n\leq x}\Lambda (n)\exp(2i\pi f(n+c_n)+\beta n)$ where $f$ a digital function, $\mathbf{c}=(c_n)$ is an almost-periodic sequence in $ \mathbb{Z}$ and $\beta $ is a real parameter, which extend the works of Mauduit-Rivat \cite{mr1} and Martin-Mauduit-Rivat \cite{mmr} to the case of the shifted prime numbers satisfying a digital constraint.

We describe the additive structure of the graded ring $\widetilde{M}_*$ of quasimodular forms over any discrete and cocompact group $\Gamma \subset \rm{PSL}(2, \RM).$ We show that this ring is never finitely generated. We calculate the exact number of new generators in each weight $k$. This number is constant for $k$ sufficiently large and equals $\dim_{\CM}(I / I \cap \widetilde{I}^2),$ where $I$ and $\widetilde{I}$ are the ideals of modular forms and quasimodular forms, respectively, of positive weight. We show that $\widetilde{M}_*$ is contained in some finitely generated ring $\widetilde{R}_*$ of meromorphic quasimodular forms with $\dim \widetilde{R}_k = O(k^2),$ i.e. the same order of growth as $\widetilde{M}_*.$, Comment: 22 pages, 1 figure