Let $S$ be an excellent extension of a (von Neumann) regular ring $R$. In this note, we study comparability of $S$ related to comparability of $R$. We show that if $R$ has the $n$-unperforation property, then $R$ satisfies $s$-comparability, almost comparability or weak comparability if and only if so does $S$.

Le but de cet article est d'établir le théorème suivant: THEOREME (HYPOTHESE DE LINDELÖF): $\zeta(\sigma + it) = O(|t|^\varepsilon)$ quel que soit le nombre réel $\varepsilon > 0$ et quel que soit $\sigma \geq \frac{1}{2}$.

We consider nonparametric minimax problem of multidimensional density estimation. Using the concept of random normalizing factor, by considering the plausible hypothesis of independence, we improve the accuracy of minimax estimation $n^{-\frac{\beta}{2\beta+d}}$: with prescribed confidence level $\alpha_{n}$, we show that the best possible attainable (random) rate is $\displaystyle{\big\{\sqrt{\log(2/\alpha_{n})}/n\big\}^{\frac{2\beta}{4\beta+d}}}$. We construct an optimal estimator and an optimal random normalizing factor in the sense of Lepski.

In this paper, admissible Mannheim partner curves are defined in pseudo-Galilean space $G_3^1$. Moreover, it is proved that the distance between the reciprocal points of admissible Mannheim curve and the torsions of these curves are constant. Furthermore, the relations for curvatures and torsions of these curves and Schell Theorem are obtained. Finally, a result about these curves being general helix is given.

New kinds of Hardy-Hilbert's integral inequalities are presented.

In this paper, we study the pair $(\mathcal {G}P(R),\mathcal {G}P(R)^{\perp})$ where $\mathcal{G}P(R)$ is the class of all Gorenstein projective modules. We prove that it is a complete hereditary cotorsion theory, provided $l. {\rm Ggldim}(R)< \infty$. We discuss also, when every Gorenstein projective module is Gorenstein flat.

Let $H$ be a real separable Hilbert space and $(a_k)_{k\in\mathbb{Z}}$ a sequence of bounded linear operators from $H$ to $H$. We consider the linear process $X$ defined for any $k$ in $\mathbb{Z}$ by $X_k=\sum_{j\in\mathbb{Z}}a_j(\varepsilon_{k-j})$ where $(\varepsilon_k)_{k\in\mathbb{Z}}$ is a sequence of i.i.d. centered $H$-valued random variables. We investigate the rate of convergence in the CLT for $X$ and in particular we obtain the usual Berry-Esseen's bound provided that $\sum_{j\in\mathbb{Z}}\vert j\vert\|a_j\|_{{\mathcal L}(H)}< +\infty$ and $\varepsilon_0$ belongs to $L_H^{\infty}$.

On étudie la dérivation de l'algèbre de Lie des champs de vecteurs polynomiaux sur $\mathbb{R}^{n}$ qui contient tous les champs constants et le champ d'Euler. Elle est adjointe de son normalisateur sur les champs de vecteurs polynomiaux de $\mathbb{R}^{n}$. Si de plus, l'algèbre de Lie contient tous les champs linéaires diagonaux alors toutes ses dérivations sont intérieures. On donne une classification de cette algèbre de Lie.

In this paper, we give a complete description of the invariant surfaces of the system governing the motion of the coupled nonlinear Schrödinger equations and their completion into abelian surfaces. We derive the associated Riemann surface on the basis of Painlevé-type analysis in the form of a genus $3$ Riemann surface $\Gamma $, which is a double ramified covering of an elliptic curve $\Gamma _{0}$ and a two sheeted genus two hyperelliptic Riemann surface $C$. We show that the affine surface $V_{c}$ obtained by setting the two quartics invariants of the problem equal to generic constants, is the affine part of an abelian surface $\widetilde{V}_{c}.$ The latter can be identified as the dual of the Prym variety ${\rm Pr} ym (\Gamma /\Gamma _{0})$ on which the problem linearizes, that is to say their solutions can be expressed in terms of abelian integrals. Also, we discuss a connection between $\widetilde{V}_{c}$ and the jacobian variety $Jac(C)$ of the genus $2$ hyperelliptic Riemann surface $C$.

The purpose of this paper is to derive some subordination and superordination results for analytic functions involving modified Saigo fractional integral operator in the open unit disk.

In this paper, we present a mathematical model of a spatial fishery with two patches. The first patch is unpolluted while fishes are infected by some toxicants released by industrial wastes in the other one. The fish stock can move between these zones, on which they are harvested by fishing fleets. Fish movements between the zones, as well as vessels displacements, are assumed to take place at a faster time scale than the variation of the fish stock and the change of the fleet size. We take advantage of these two time scales to derive a reduced model governing the dynamics of the total fish stock and fishing effort. This reduced model is analyzed and we look for existence of limit cycle. We also present numerical simulations to illustrate the results.

On étudie le premier espace de cohomologie de Chevalley-Eilenberg des algèbres de Lie attachées à une distribution non régulière involutive d'une variété différentiable. On applique les résultats obtenus à l'algèbre de Lie des champs de vecteurs à support compact et aux algèbres de Lie relatives à un feuilletage généralisé, conduisant à une généralisation d'un théorème de Kanie et de certains résultats de Lichnérowicz.

By introducing some parameters and estimating the weight coefficient, we establish a new Hilbert's type inequality with a best constant factor. As applications, the reverse form, some particular results and the equivalent form are considered.