Chaotic systems confer distinct advantages for image encryption due to their remarkable responsiveness to initial parameters, inherent unpredictability, and characteristics of pseudo-randomness. However, many existing chaos-based image encryption schemes still suffer from limitations such as narrow chaotic parameter intervals, insufficient trajectory uniformity, and inadequate diffusion depth, which may weaken the randomness of generated sequences and reduce encryption robustness. To address these issues, a square-term enhanced four-dimensional chaotic system was constructed, and a color image encryption scheme integrating hierarchical scrambling and symmetric bidirectional diffusion was developed. Dynamic analyses, including phase portraits, bifurcation diagrams, maximum Lyapunov exponents, time-series distribution, and complexity evaluation, indicate that the proposed system exhibits a broad chaotic interval, strong initial-value sensitivity, and improved trajectory uniformity. It can well-adapt to the image encryption requirements of different categories. Simulation experiments conducted using a variety of test images indicate that the encryption scheme offers an adequately large key space, with pixel correlation measured at less than 0.03, information entropy exceeding 7.9974, and robust resistance to differential attacks. The computed mean values for number of pixels change rate (NPCR) and unified average changing intensity (UACI) were found to be 99.6073% and 33.4626%, respectively, demonstrating a strong alignment with the optimal benchmarks. The algorithm achieves competitive performance in terms of efficiency and security, can effectively resist common attacks, and is suitable for the secure transmission and storage of sensitive images.

A two-step randomized Gauss-Seidel (TRGS) method is presented for large linear least squares problem with tall and narrow coefficient matrix. The TRGS method projects the approximate solution onto the solution space by given two random columns and is proved to be convergent when the coefficient matrix is of full rank. Several numerical examples show the effectiveness of the TRGS method among all methods compared.

Let $(X,d)$ be a metric space and $\mathcal{H}(X)$ represent all non-empty, compact subsets of $X$. The expansivity of the multivalued map sequence ${\overline{f}}_{1,\mathrm{\infty }}:\mathcal{H}(X)\to \mathcal{H}(X)$, including expansivity, positive ${\mathrm{\aleph }}_0$-expansivity, were investigated. Also, stronger forms of sensitivities, such as multi-sensitivity and syndetical sensitivity, were explored. This research demonstrated that some chaotic properties can be mutually derived between $(f_{1,\mathrm{\infty }},X)$ and $({\overline{f}}_{1,\mathrm{\infty }},\mathcal{H}(X))$, showing fundamental similarities between these systems. Conversely, the inability to derive other properties underlined essential differences between them. These insights are crucial for simplifying theoretical models and enhancing independent research. Lastly, the relationship between expansivity and sensitivity was discussed and the concept of topological conjugacy to the system $({\overline{f}}_{1,\mathrm{\infty }},\mathcal{H}(X))$ was extended.