It is not completely clear which elements constitute the frame sets of the B-splines currently, but some considerable results have been obtained. In this paper, firstly, the background of frame set is introduced. Secondly, the main progress of the frame sets of the B-splines in the past more than twenty years are reviewed, and particularly the progress for the frame set of the 2 order B-spline and the frame set of the 3 order B-spline are explained, respectively.

In this note, we consider a class of Fourier integral operators with rough amplitudes and rough phases. When the index of symbols in some range, we prove that they are bounded on L^1 and construct an example to show that this result is sharp in some cases. This result is a generalization of the corresponding theorems of Kenig-Staubach and Dos Santos Ferreira-Staubach.

Given a connected graph G, the revised edge-revised Szeged index is defined as S{z}_e^{\ast }\left(G\right)=\sum _{e=uv\in E_G}\left(m_u\right(e)+\frac{m_0\left(e\right)}{2})\left(m_v\right(e)+\frac{m_0\left(e\right)}{2}), where m_u\left(e\right), m_v\left(e\right) and m_0\left(e\right) are the number of edges of G lying closer to vertex u than to vertex v, the number of edges of G lying closer to vertex v than to vertex u and the number of edges of G at the same distance to u and v, respectively. In this paper, by transformation and calculation, the lower bound of revised edge-Szeged index of unicyclic graphs with given diameter is obtained, and the extremal graph is depicted.

As the extension of classical Hardy operator and Cesàro operator, Hausdorff operator plays an important role in the harmonic analysis, so it is significant to discuss the boundedness of this kind of operator on various function spaces. The article explores the boundedness of a kind of Hausdorff operators on Lebesgue spaces and calculates the optimal constants for the operators to be bounded on such spaces. In addition, the paper also obtains the necessary and sufficient for a kind of multilinear Hausdorff operators to be bounded on Lebesgue spaces and their optimal constants.

In this paper, a three-term derivative-free projection method is proposed for solving nonlinear monotone equations. Under some appropriate conditions, the global convergence and R-linear convergence rate of the proposed method are analyzed and proved. With no need of any derivative information, the proposed method is able to solve large-scale nonlinear monotone equations. Numerical comparisons show that the proposed method is effective.