This paper mainly focuses on solitary waves excited by topography with time-dependent variable coefficient. By making use of multiple scale expansion and multiple level approximation method, a variable coefficient KdV equation with variable coefficient topographic forcing term is derived from barotropic and potential vorticity equation on a beta-plane including topography effect. In the derivation, removing y-average trick, a higher order term of stream function including five arbitrary functions and forced topography is introduced. Taking the strict solution of the standard constant coefficient KdV equation as the initial value, the approximate analytical solution of the derived equation is obtained by means of homotopy analysis method. Based on the new equation and its analytical solution, some complicated and changeable atmospheric blocking phenomena might be explained when some functions are selected appropriately.

BACKGROUND: Periodontitis is a common oral immune inflammatory disease and early detection plays an important role in its prevention and progression. However, there are no accurate biomarkers for early diagnosis. OBJECTIVE: This study screened periodontitis-related diagnostic biomarkers based on weighted gene correlation network analysis and machine algorithms. METHODS: Transcriptome data and sample information of periodontitis and normal samples were obtained from the Gene Expression Omnibus (GEO) database, and key genes of disease-related modules were obtained by bioinformatics. The key genes were subjected to Gene Ontology (GO) enrichment analysis, Kyoto Encyclopedia of Genes and Genomes (KEGG) enrichment analysis and 5 machine algorithms: Logistic Regression (LR), Random Forest (RF), Gradient Boosting Decisio Tree (GBDT), Extreme Gradient Boosting (XGBoost), and Support Vector Machine (SVM). Expression and correlation analysis were performed after screening the optimal model and diagnostic biomarkers. RESULTS: A total of 47 candidate genes were obtained, and the LR model had the best diagnostic efficiency. The COL15A1, ICAM2, SLC15A2, and PIP5K1B were diagnostic biomarkers for periodontitis, and all of which were upregulated in periodontitis samples. In addition, the high expression of periodontitis biomarkers promotes positive function with immune cells. CONCLUSION: COL15A1, ICAM2, SLC15A2 and PIP5K1B are potential diagnostic biomarkers of periodontitis.

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A table lookup method for solving nonlinear fractional partial differential equations (fPDEs) is proposed in this paper. Looking up the corresponding tables, we can quickly obtain the exact analytical solutions of fPDEs by using this method. To illustrate the validity of the method, we apply it to construct the exact analytical solutions of four nonlinear fPDEs, namely, the time fractional simplified MCH equation, the space-time fractional combined KdV-mKdV equation, the (2+1)-dimensional time fractional Zoomeron equation, and the space-time fractional ZKBBM equation. As a result, many new types of exact analytical solutions are obtained including triangular periodic solution, hyperbolic function solution, singular solution, multiple solitary wave solution, and Jacobi elliptic function solution.