The alternating links give a classical class of links. They play an important role in Knot Theory. Ozsváth and Szabó introduced a quasi-alternating link which is a generalization of an alternating link. In this paper we review some results of alternating links and quasi-alternating links on the Jones polynomial and the Khovanov homology. Moreover, we introduce a long pass link. Several problems worthy of further study are provided.

For an irreducible character \chi of a finite group G, we define its codegree as \mathrm{c}\mathrm{o}\mathrm{d}\left(\chi \right)=\frac{|G:\ker \chi |}{\chi \left(1\right)}. In this paper, we introduce some known results and unsolved problems about character codegrees in finite groups.

A generalized strongly regular graph of grade p, as a new generalization of strongly regular graphs, is a regular graph such that the number of common neighbours of both any two adjacent vertices and any two non-adjacent vertices takes on p distinct values. For any vertex v of a generalized strongly regular graph of grade 2 with parameters (n,k\mathrm{;}{a}_1,a_2\mathrm{;}{c}_1,c_2), if the number of the vertices that are adjacent to v and share a_i(i=1,2) common neighbours with v, or are non-adjacent to v and share c_i(i=1,2) common neighbours with v is independent of the choice of the vertex v, then the generalized strongly regular graph of grade 2 is free. In this paper, we investigate the generalized strongly regular graph of grade 2 with parameters (n,k\mathrm{;}k-1,a_2\mathrm{;}k-1,c_2) and provide the sufficient and necessary conditions for the existence of a family of free generalized strongly regular graphs of grade 2.

The existence of bounded weak solutions, to a class of nonlinear elliptic equations with variable exponents, is investigated in this article. A uniform a priori L^{\mathrm{\infty }} estimate is obtained by the De Giorgi iterative technique. Thanks to the weak convergence method and Minty's trick, the existence result is proved through limit process.

We propose a class of new hierarchical model for the evolution of two interacting age-structured populations, which is a system of integro-partial differential equations with global feedback boundary conditions and may describe the interactions such as competition, cooperation and predator-prey relation. Based upon a group of natural conditions, the existence and uniqueness of solutions on infinite time interval are proved by means of fixed point and extension principle, and the continuous dependence of the solution on the initial age distribution is established. These results lay a sound basis for the investigation of stability, controllability and variable optimal control problems.

In this paper, we study the continuous dependence of eigenvalue of Sturm-Liouville differential operators on the boundary condition by using of implicit function theorem. The work not only provides a new and elementary proof of the above results, but also explicitly presents the expressions for derivatives of the n-th eigenvalue with respect to given parameters. Furthermore, we obtain the new results of the position and number of the generated double eigenvalues under the real coupled boundary condition.