The position vectors of regular rectifying curves always lie in their rectifying planes. These curves were well investigated by B.Y.Chen. In this paper, the concept of framed rectifying curves is introduced, which may have singular points. We investigate the properties of framed rectifying curves and give a method for constructing framed rectifying curves. In addition, we reveal the relationships between framed rectifying curves and some special curves.

The authors of this paper prove the existence and regularity results for the homogeneous Dirichlet boundary value problem to the equation − div ( | ∇ u | p − 2 ∇ u ) = f ( x ) u α ( x ) $-\operatorname{div}(|\nabla u|^{p-2}\nabla u)=\frac{f(x)}{u ^{\alpha(x)}}$ with f ∈ L m ( Ω ) $f\in L^{m}(\Omega)$ ( m ⩾ 1 $m\geqslant1$ ) and α ( x ) > 0 $\alpha(x)>0$ . Due to the nonlinearity of a p-Laplace operator and the anisotropic variable exponent α ( x ) $\alpha(x)$ , some classical methods may not directly be applied to our problem. In this paper, we construct a suitable test function and apply the Leray-Schauder fixed point theorem to prove the existence of positive solutions with necessary a priori estimate and compact argument. Furthermore, we also discuss the relationship among the regularity of solutions, the summability of f and the value of α ( x ) $\alpha(x)$ .

In this paper, we define the supersolvable order of hyperplanes in a supersolvable arrangement, and obtain a class of inductively free arrangements according to this order. Our main results improve the conclusion that every supersolvable arrangement is inductively free. In addition, we assert that the inductively free arrangement with the required induction table is supersolvable.

The purpose of this paper is twofold. Firstly, we generalize the notion of semigeneric braid arrangement to semigeneric graphical arrangement. Secondly, we give a formula for the characteristic polynomial of the semigeneric graphical arrangement $ \mathcal{S}_G: = \{x_i- x_j = a_i: ij\in E(G), 1\leq i < j\leq n \} $, where $ a_i{\rm{'s}} $ are generic. The formula is obtained by characterizing central subarrangements of $ \mathcal{S}_{G} $ via the corresponding graph $ {G} $.