In this paper, several necessary and sufficient conditions are presented for a module $M$ to be cohomologically complete intersection module with respect to $I$, i.e. $H_I^i(M)=0$ for all $i \not= c = \mathrm{grade}(I, M)$. This notion is a generalization of cohomologically complete intersection ideals.

Our aim in this paper is to deal with the boundedness of the Hardy- Littlewood maximal operator and the Hardy operator on non-homogeneous central Herz-Morrey-Musielak-Orlicz spaces and to establish a generalization of Sobolev’s inequalities for Riesz potentials of functions in such spaces.

Given a planar Jordan domain $G$ with rectifiable boundary, it is well known that smooth functions on the closure of $G$ do not always admit smooth extensions to $\mathbb C$. Further conditions on the boundary are necessary to guarantee such extensions. On the other hand, Weierstrass’ approximation theorem yields polynomials converging uniformly to $f \in C(\overline{G}, \mathbb C)$. In this note we show that for Vitushkin sets $K$ with $K = \overline{K^\circ}$ it is always possible to uniformly approximate on $K$ the smooth function $f \in C^1(K, \mathbb C)$ by smooth functions $f_n$ in $\mathbb C$ so that also $\overline{\partial}f_n$ converges uniformly to $\overline{\partial}f$ on $K$. As a byproduct we deduce from its ‘‘smooth in a neighborhood version’’ the general Gauss integral theorem for functions whose partial derivatives in $G$ merely admit continuous extensions to its boundary.

A distance-squared function is one of the most significant functions in the application of singularity theory to differential geometry. Moreover, distance-squared mappings are naturally extended mappings of distance-squared functions, wherein each component is a distance-squared function. In this paper, compositions of a given plane curve and generic distance-squared mappings on the plane into the plane are investigated from the viewpoint of stability.

The notion of S-stability of foliations on branched simple polyhedrons is introduced by R. Benedetti and C. Petronio in the study of characteristic foliations of contact structures on 3-manifolds. We additionally assume that the 1-form $\beta$ defining a foliation on a branched simple polyhedron $P$ satisfies $d\beta > 0$, which means that the foliation is possibly a characteristic foliation of a contact form whose Reeb flow is transverse to $P$. In this paper, we show that if there exists a 1-form $\beta$ on $P$ with $d\beta > 0$ then we can find a 1-form with the same property and additionally being S-stable. We then prove that the number of simple tangency points of an S-stable foliation on a positive or negative flow-spine is at least 2 and give a recipe for constructing a characteristic foliation of a 1-form $\beta$ with $d\beta > 0$ on the abalone.

It is well-known that the second highest coefficient of the Alexander polynomial of any lens space knot in $S^3$ is −1. We show that if the third highest coefficient of the Alexander polynomial $\Delta_K(t)$ of a lens space knot $K$ in $S^3$ is non-zero, then $\Delta_K(t)$ coincides with the Alexander polynomial of the $(2,2g+1)$-torus knot, where $g$ is the Seifert genus of $K$.