We prove an alternative method in order to obtain generation and analyticity results for the semigroups generated by some degenerate second order differential operators linked to the hypergeometric equation.

Lie quasi-bialgebras are natural generalisations of Lie bialgebras introduced by Drinfeld. To any finite-dimensional Lie quasi-bialgebra structure $(\mathcal{G}, \mu, \gamma, \phi)$ and a $\mathcal{D}$-module structure $M$, where $\mathcal{D}$ is the double of the given Lie quasi-bialgebra, we associate one operator $L_{M} =\partial_{\mu, M}d_{\gamma, M} + d_{\gamma, M}\partial_{\mu, M}$ called the laplacien of the Lie quasi-bialgebra associated to the $\mathcal{D}$-module structure. We establish the fondamentals properties of the laplacian and give an explicit formula for $L_{M}$ by mean of adjoint characters of $\mathcal{G}$ and $\mathcal{G^*}$.

We obtain the expression of Ricci tensor for a $GCR$-lightlike submanifold of indefinite complex space form and discuss the properties of Ricci tensor on totally geodesic $GCR$-lightlike submanifold of an indefinite complex space form.

In this paper, we discuss $A$-modules and $L$-modules (resp. $L$-comodules) for Hom-Lie algebras (resp. Hom-Lie coalgebras). We show that for a given Hom-associative algebra $A$ (resp. Hom-coassociative coalgebra), the $A$-module (resp. comodule) extends to $L(A)$-module (resp. comodule), where $L(A)$ is the associated Lie algebra (resp. Lie coalgebra), with the same structure map. We also prove that $L$-modules become $L_\alpha$-modules, where $L_\alpha$ is the Hom-Lie algebra obtained from the Lie algebra $L$ by stwisting the Lie bracket. Then we introduce Hom-Lie quasi-bialgebras and prove that a Lie quasi-bialgebra turns to a Hom-Lie quasi-bialgebra by stwisting the Lie quasi-bialgebra structure by an endomorphism. Moreover, we show that an exact Lie quasi-bialgebra extends to an exact Hom-Lie quasi-bialgebra.

This note completes [15]. Sharp results on Poisson summation formula and irregular sampling are announced. In both cases tools from [15] are needed.

In this paper we study in quantum calculus the theory of inverse problem and approximation in a large class of Hilbert spaces with reproducing kernels.

Let $X$ be a simply connected CW complex of finite rational LS-category with $\dim H_n(X;{\mathbb Q})<\infty$ for all $n$. The dimension of rational Gottlieb group $G_*(X)\otimes {\mathbb Q}$ is upper-bounded by the rational LS-category $cat_0(X)$ the inequation $\dim G_*(X)\otimes {\mathbb Q}\leq cat_0(X)$ holds [2]. Then we introduce a new rational homotopical invariant between them, denoted as the pairing rank $v_0(X)$ in the rational homotopy group $\pi_*(X)\otimes {\mathbb Q}$ such that $\dim G_*(X)_{\mathbb Q}\leq v_0(X)\leq cat_0(X)$. If $\pi_*(f)\otimes {\mathbb Q}$ is injective for a map $f:X\to Y$, then we have $v_0(X)\leq v_0(Y)$. Also it has a good estimate for a fibration $X{\to} E{\to} Y$ as $v_0(E)\leq v_0(X) +v_0(Y)$.

In this paper, $CR$-submanifolds of a nearly trans-hyperbolic Sasakian manifold with a semi-symmetric non-metric connection are studied. The parallel distributions relating to $\xi$-vertical and $\xi$-horizontal $CR$-submanifolds of nearly trans-hyperbolic Sasakian manifold with a semi symmetric non-metric connection are obtained. Moreover, Nijenhuis tensor is calculated and integrability conditions of the distributions on $CR$ submanifolds of nearly trans-hyperbolic Sasakian manifold with a semi-symmetric non-metric connection are discussed.