The aim of this paper is to study the classical global and weak dimensions of the amalgamated duplication of a ring $R$ along a pure ideal $I$.

Let $G$ be a locally compact Lie group, $\mathcal{G}$ its Lie algebra and $n$ an integer $\geq 1$. We build a locally compact Lie group, noted $G_{n}$, whose Lie algebra is the generalized Takiff algebra of order $n$ associated with $\mathcal{G}$. We investigate some properties of this group. As application, we show that $SL_{2}(\mathbb{R}^{n+1})$ is the generalized Takiff group of order $n$ associated with $SL_{2}(\mathbb{R})$, where $\mathbb{R}^{n+1}$ is equiped with an appropriate algebras structure.

Here we consider an infinite sub-family of the generalized Petersen graphs $P(n,k)$ for $n=2k+1\geq 3,$ and using the two algorithms that A. Behzad et al presented in [1], we determine an upper bound and a lower bound for the independent domination numbers of these graphs.

In this paper, we give some conditions that insure the non-negativity of the solution of the relativistic Boltzmann equation in a full curved space-time, generalizing thereby some previous results known in the flat Minkowski space-time and for the non-relativistic Boltzmann equation.

In this paper we have defined torse-forming projective $N-$ curvature collineation and discuss the existence of torse-forming projective $N-$ curvature collineation in $NP-F_{n}$ (normal projective Finsler space) and study the corresponding results for contra, concurrent and special concircular transformations.

In this paper, $K$ denotes a complete, non-trivially valued, non-archimedean field. Sequences and infinite matrices have entries in $K$. Supplementing [4], we make a further study of sequences of 0's and 1's in $K$.

We are interested in integral functionals of the form \begin{equation*} \boldsymbol{J}(U, V) =\int_{\Omega }J\big(x, U(x), V(x)\big) dx, \end{equation*} where $J$ is Carathéodory positive integrand, satisfying some growth condition of order $p\in]1, +\infty[$. We show that $\mathcal{A}(x, \partial)-$quasiconvexity of the integrand $J$ with respect to the third variable is a necessary and sufficient condition of lower semicontinuity of $\boldsymbol{J}$, where $\mathcal{A}(x, \partial)$ is a differential operator given by \begin{equation*} \mathcal{A}(x, \partial)=\sum_{j=1}^{N}A^{(j)}(x)\partial_{x_{j}}, \end{equation*} and the coefficients $A^{(j)}, j=1,...,N$ are only Lipschitzian, i.e. $A^{(j)}\in W^{1,\infty }\big(\Omega; \mathbb{M}^{l\times d}\big)$ and satisfy the condition of constant rank. To this end, a framework of paradifferential calculus is needed to deal with the lower smoothness of the coefficients.

This paper was retracted by the publisher as it was found that K. Ratchagit plagiarized a paper by Vu Ngoc Phat and Phan T. Nam published in the Electonic Journal of Differential Equations, vol. 2005 (2005), no. 58, pp. 1–8.

L'objet de cet article est d'établir les résultats suivants :

Théorème (I) (CONJECTURE JACOBIENNE)

Soient $K$ un corps de caractéristique $0,\: A = K [T_1,\dots, T_n]$ l'anneau des polynômes à $n$ variables sur $K$ et $f_1,\dots, f_n$ $n$ éléments de $A$. On suppose que $\mathrm{d\acute{e}t} (\partial f_i/\partial T_j)$ est une constante non nulle de $K.$ Alors l'injection canonique $K [f_1,\dots, f_n] \longrightarrow K[T_1,\dots, T_n]$ est un isomorphisme.

Théorème (II) (CONJECTURE JACOBIENNE GENERALISEE)

Soient $R$ un anneau intègre, $A = R[T_1,\dots, T_n]$ l'anneau des polynômes à $n$ variables sur $R$, et $f_1,\dots, f_n$ $n$ éléments de $A$. On suppose que $\mathrm{d\acute{e}t}(\partial \ f_i/\partial T_j)$ est un élément inversible de $R$. Alors les conditions suivantes sont équivalentes :

1) L'injection canonique $R[f_1,\dots, f_n] \longrightarrow R[T_1,\dots, T_n]$ est un isomorphisme.

2) Le degré de l'extension du corps des fractions de $A = R[T_1,\dots, T_n]$ sur celui de $B = R[f_1,\dots, f_n]$ n'est pas divisible par la caractéristique du corps des fractions de $R$.