Life activities are extremely complex phenomena in nature. From molecular signaling regulation to multi-cellular tissue formation and so on, the biological system consists of multiple temporal, spatial and functional scales. Multiscale mathematical models have extensive applications in life science research due to their capacity of appropriately simulating the complex multiscale biological systems. Many mathematical methods, including deterministic methods, stochastic methods as well as discrete or rule-based methods, have been widely used for modeling biological systems. However, the models at single scale are not sufficient to simulate complex biological systems. Therefore, in this paper we give a survey of two multiscale modeling approaches for biological systems. One approach is continuous stochastic method that couples ordinary differential equations and stochastic differential equations; Another approach is hybrid continuous-discrete method that couples agent-based model with partial differential equations. We then introduce the applications of these multiscale modeling approaches in systems biology and look ahead to the future research.

In this paper, path-connectivity of the set of some special wavelets in L^2\left(\mathbb{R}\right), which is the topological geometric property of wavelets, is introduced. In particular, the main progress of wavelet connectivity in the past twenty years is reviewed and some unsolved problems are listed. The corresponding results of high dimension case and other cases are also briefly explained.

We are concerned with the numerical methods for nonlinear equation and their semilocal convergence in this paper. The construction techniques of iterative methods are induced by using linear approximation, integral interpolation, Adomian series decomposition, Taylor expansion, multi-step iteration, etc. The convergent conditions and proof methods, including majorizing sequences and recurrence relations, in semilocal convergence of iterative methods for nonlinear equations are discussed in the theoretical analysis. The majorizing functions, which are used in majorizing sequences, are also discussed in this paper.

The iterated spherical average \mathrm{\Delta }(A_1{)}^N is an important operator in harmonic analysis, and has very important applications in approximation theory and probability theory, where \mathrm{\Delta } is the Laplacian, A_1 is the unit spherical average and (A_1{)}^N is its iteration. In this paper, we mainly study the sufficient and necessary conditions for the boundedness of this operator in Besov-Lipschitz space, and prove the boundedness of the operator in Triebel-Lizorkin space. Moreover, we use above conclusions to improve the existing results of the boundedness of this operator in L^p space.

Let k⩾1 be an integer. Assume that RH holds. In this paper we prove that a suitable asymptotic formula for the average number of representations of integers n=p_1^k+p_2^3+p_3^3+p_4^3+p_5^3, where p_1,p_2,p_3,p_4,p_5 are prime numbers. This expands the previous results.