As liver hepatocellular carcinoma (LIHC) has high morbidity and mortality rates, improving the clinical diagnosis and treatment of LIHC is an important issue. The advent of the era of precision medicine provides us with new opportunities to cure cancers, including the accumulation of multi-omics data of cancers. Here, we proposed an integration method that involved the Fisher ratio, Spearman correlation coefficient, classified information index, and an ensemble of decision trees (DTs) for biomarker identification based on an unbalanced dataset of LIHC. Then, we obtained 34 differentially expressed genes (DEGs). The ability of the 34 DEGs to discriminate tumor samples from normal samples was evaluated by classification, and a high area under the curve (AUC) was achieved in our studied dataset and in two external validation datasets (AUC = 0.997, 0.973, and 0.949, respectively). Additionally, we also found three subtypes of LIHC, and revealed different biological mechanisms behind the three subtypes. Mutation enrichment analysis showed that subtype 3 had many enriched mutations, including tumor protein p53 (TP53) mutations. Overall, our study suggested that the 34 DEGs could serve as diagnostic biomarkers, and the three subtypes could help with precise treatment for LIHC.

The Mahalanobis–Taguchi system (MTS) is a multivariate data diagnosis and prediction technology, which is widely used to optimize large sample data or unbalanced data, but it is rarely used for high-dimensional small sample data. In this paper, the optimized MTS for the classification of high-dimensional small sample data is discussed from two aspects, namely, the inverse matrix instability of the covariance matrix and the instability of feature selection. Firstly, based on regularization and smoothing techniques, this paper proposes a modified Mahalanobis metric to calculate the Mahalanobis distance, which is aimed at reducing the influence of the inverse matrix instability under small sample conditions. Secondly, the minimum redundancy-maximum relevance (mRMR) algorithm is introduced into the MTS for the instability problem of feature selection. By using the mRMR algorithm and signal-to-noise ratio (SNR), a two-stage feature selection method is proposed: the mRMR algorithm is first used to remove noise and redundant variables; the orthogonal table and SNR are then used to screen the combination of variables that make great contribution to classification. Then, the feasibility and simplicity of the optimized MTS are shown in five datasets from the UCI database. The Mahalanobis distance based on regularization and smoothing techniques (RS-MD) is more robust than the traditional Mahalanobis distance. The two-stage feature selection method improves the effectiveness of feature selection for MTS. Finally, the optimized MTS is applied to email classification of the Spambase dataset. The results show that the optimized MTS outperforms the classical MTS and the other 3 machine learning algorithms.

In this paper, we study the following three species predator-prey model with prey-taxis: * \begin{document}$ \left\{ \begin{array}{lll} u_t = d_1\Delta u+\chi_1\nabla\cdot(u\nabla v)+r_1u(1-u-kv-b_1w), &\quad x\in \Omega, t>0, \\ v_t = d_2\Delta v+r_2v(1-hu-v-b_2w), &\quad x\in \Omega, t>0, \\ w_t = d_3\Delta w-\chi_2\nabla\cdot(w\nabla u)-\chi_3\nabla\cdot(w\nabla v)\\ \ \ \ \ \ \ \ +r_3w(-1+au+av-w), &\quad x\in \Omega, t>0. \end{array}\right. $\end{document} We prove that if (1.7) and (1.6) hold, the model (\begin{document}$ \ast $\end{document}) admits at least one global generalized solution in any dimension.