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Featured Application: Privacy protection and data mining. This study aims to improve the quality of the clustering results of the density peak clustering (DPC) algorithm and address the privacy protection problem in the clustering analysis process. To achieve this, a DPC algorithm based on Chebyshev inequality and differential privacy (DP-CDPC) is proposed. Firstly, the distance matrix is calculated using cosine distance instead of Euclidean distance when dealing with high-dimensional datasets, and the truncation distance is automatically calculated using the dichotomy method. Secondly, to solve the difficulty in selecting suitable clustering centers in the DPC algorithm, statistical constraints are constructed from the perspective of the decision graph using Chebyshev inequality, and the selection of clustering centers is achieved by adjusting the constraint parameters. Finally, to address the privacy leakage problem in the cluster analysis, the Laplace mechanism is applied to introduce noise to the local density in the process of cluster analysis, enabling the privacy protection of the algorithm. The experimental results demonstrate that the DP-CDPC algorithm can effectively select the clustering centers, improve the quality of clustering results, and provide good privacy protection performance. [ABSTRACT FROM AUTHOR]

Using elementary methods, we obtain an explicit formula for the fourth power meanΣm=1q'Σχmodq|Σa=1q'χ(a)e(mak+naq)|4 for arbitrary positive integer k, where e(y)=e2πiy, χ is a Dirichlet character modulo q and Σa=1q' denotes the summation over all a such that (a,q)=1. This improves H.N. Liu's result by averting the restriction (k,q)=1. [ABSTRACT FROM AUTHOR]