To obtain chaotic systems with good properties to help chaos based image encryption, this paper first study the condition under which quadratic functions can be used as pseudo-random sequence generator. We give a necessary and sufficient condition for quadratic functions topologically conjugate with the most frequently used Logistic map and Tent map. Since topological conjugacy maintains chaotic characteristics, more quadratic functions can be used as pseudo-random sequence generators in the digital image encryption. Then, it analyzes a diffusion algorithm with a good effect but poor in resisting the chosen plaintext attack. It is also verified by simulation. The quadratic function chaotic system is applied in the improvement of the diffusion algorithm. Furthermore, the improved algorithm contains plaintext related and cyclic shift operations. Finally, the performance analysis shows that the improved scheme has good security and high efficiency.

Topological data analysis has been acknowledged as one of the most successful mathematical data analytic methodologies in many fields. Additionally, it has also been gradually applied in financial time series analysis and proved effective in exploring the topological features of such data. We select 100 stocks from China’s markets and construct point cloud data for topological data analysis. We detect critical dates from the L^p-norms of the persistence landscapes. Our results reveal the dates are highly consistent with the transition time of some major events in the sample period. We compare the correlations and statistical properties of stocks before and during the events via complex networks to describe the markets’ situation. The strength and variation of links among stocks are clearly different during the major events. We also investigate the neighborhood features of stocks from topological perspectives. This helps identify the important stocks and explore their situations under each event. Finally, we cluster the stocks based on the neighborhood features, which exhibit the heterogeneity impact on stocks of the different events. Our work demonstrates that topological data analysis has strong applicability in the dynamic correlations of stocks.

This paper investigates the robust stability and stabilization of networked evolutionary games (NEGs) with time delays. First, a mathematical model is presented to describe the dynamics of NEG with time-varying delays and disturbances. Second, an auxiliary system is constructed using the semi-tensor product of matrices and a dimension augmenting technique. Then, a verification condition of robust stability is derived. Third, in order to stabilize NEG to the Nash equilibrium, the robust stability problem is transformed into the robust stabilization problem. Moreover, an algorithm is proposed to design the stabilization controller. Finally, the validity of the results is verified by an example.

The purpose of this paper is to provide a Jacobi-collocation method for solving second kind Volterra integral equations with a smooth kernel. This method leads to a fully discrete integral operator. First, it is shown that the fully discrete integral operator is stable in bothL∞and weightedL2norms. Then, the proposed approach is proved to arrive at an optimal (the most possible) convergent order in both norms. One numerical example demonstrates the efficiency and accuracy of the proposed method.

Since the 1930s, topological counting on finite sets has been an interesting work so as to enumerate the number of corresponding order relations on the sets. Starting from the semi-tensor product (STP), we give the expression of the relationship between subsets of finite sets from the perspective of algebra. Firstly, using the STP of matrices, we present the algebraic representation of the subset and complement of finite sets and corresponding structure matrices. Then, we investigate respectively the relationship between the intersection and union and intersection and minus of structure matrices. Finally, we provide an algorithm to enumerate the numbers of topologies on a finite set based on the above theorems.

Some results are obtained on finite unions ofD-spaces. It is proved that if a space is the union of finitely many locally compactD-subspaces, then it is aD-space. It follows that a space is aD-space if it is the union of finitely many locally compact submetacompact subspaces. And a space is aD-space if it is the union of aD-subspace with a locally compactD-subspace. This partially answers one problem raised by Arhangel’skii. At last, some examples are given to exhibit the applications of nearly good relation to discoverD-classes.