The recession cone and recession function are very important research objects in Convex Analysis. They have extensive applications in the optimization theory. Firstly, we study the properties of the recession cone and recession function. The positive homogeneity and subadditivity of recession function are mainly discussed. And the different methods are considered to prove these properties. Secondly, we discuss the unboundedness of the convex sets and convex functions by using recession cone and recession function.

Usually in the study of limit problems, will encounter more complex problems, in this paper, we discuss how to use the concept of equivalent infinitesimal better limit operation. At the same time, in the process of research, we re-explore the proof of Taylor's formula, and find that some functions have a similar expansion form to Taylor's formula, that is, 'fractional expansion'. It is also found that after the linear combination of Taylor expansion and fractional expansion, the obtained expansion is more accurate, which helps us to have a better understanding of the approximation of function expansion.

This study, through the method of canonical correlation analysis, revealed significant correlations between various dimensions of learning attitudes of students and various dimensions of teacher knowledge. An analysis of data from a group of 221 high school students showed that teacher knowledge of teaching content, theoretical knowledge, and teaching practice and classroom management significantly impact learning attitudes of students. Specifically, teacher knowledge of teaching content plays a crucial role in promoting students’ behavioral inclination to learn chemistry, teachers’ theoretical knowledge significantly enhances students’ liking for chemistry laboratory courses, while teachers’ teaching practice and classroom management have a suppressive effect on students’ evaluative beliefs about school chemistry. The results of this study provide effective guidance for both the theory and practice of high school chemistry education.

This paper presents a numerical method for solving a nonlinear age-structured population model based on a set of piecewise constant orthogonal functions. The block-pulse functions (BPFs) method is applied to determine the numerical solution of a non-classic type of partial differential equation with an integral boundary condition. BPFs duo to the simple structure can efficiently approximate the solution of systems with local or non-local boundary conditions. Numerical results reveal the accuracy of the proposed method even for the long term simulations.