Regular solutions of the stationary Navier-Stokes equations on high dimensional Euclidean space

We study the existence of regular solutions of the incompressible stationary Navier-Stokes equations in $n$-dimensional Euclidean space with a given bounded external force of compact support. In dimensions $n\le 5$, the existence of such solutions was known. In this paper, we extend it to dimensions $n\le 15$.Comment: Exposition improved. To appear in Comm. Math. Phy

Asymptotics of the solution to the perfect conductivity problem with pp-Laplacian

We study the perfect conductivity problem with closely spaced perfect conductors embedded in a homogeneous matrix where the current-electric field relation is the power law $J=\sigma|E|^{p-2}E$. The gradient of solutions may be arbitrarily large as $\varepsilon$, the distance between inclusions, approaches to 0. To characterize this singular behavior of the gradient in the narrow region between two inclusions, we capture the leading order term of the gradient. This is the first gradient asymptotics result on the nonlinear perfect conductivity problem.Comment: 37 pages, 1 figur

Comprehensive characterization of endoplasmic reticulum stress in bladder cancer revealing the association with tumor immune microenvironment and prognosis

Background: This study constructs a molecular subtype and prognostic model of bladder cancer (BLCA) through endoplasmic reticulum stress (ERS) related genes, thus helping to clinically guide accurate treatment and prognostic assessment.Methods: The Bladder Cancer (BLCA) gene expression data was downloaded from The Cancer Genome Atlas (TCGA) and Gene Expression Omnibus (GEO) database. We clustered by ERS-related genes which obtained through GeneCards database, results in the establishment of a new molecular typing of bladder cancer. Further, we explored the characteristics of each typology in terms of immune microenvironment, mutations, and drug screening. By analyzing the ERS-related genes with univariate Cox, LASSO and multivariate Cox analyses, we also developed the four-gene signature, while validating the prognostic effect of the model in GSE32894 and GSE13507 cohorts. Finally, we evaluated the prognostic value of the clinical data in the high and low ERS score groups and constructed a prognostic score line graph by Nomogram.Results: We constructed four molecular subtypes (C1- C4) of bladder cancer, in which patients with C2 had a poor prognosis and those with C3 had a better prognosis. The C2 had a high degree of TP53 mutation, significant immune cell infiltration and high immune score. In contrast, C3 had a high degree of FGFR3 mutation, insignificant immune cell infiltration, and reduced immune checkpoint expression. After that, we built ERS-related risk signature to calculate ERS score, including ATP2A3, STIM2, VWF and P4HB. In the GSE32894 and GSE13507, the signature also had good predictive value for prognosis. In addition, ERS scores were shown to correlate well with various clinical features. Finally, we correlated the ERS clusters and ERS score. Patients with high ERS score were more likely to have the C2 phenotype, while patients with low ERS score were C3.Conclusion: In summary, we identified four novel molecular subtypes of BLCA by ERS-related genes which could provide some new insights into precision medicine. Prognostic models constructed from ERS-related genes can be used to predict clinical outcomes. Our study contributes to the study of personalized treatment and mechanisms of BLCA

The insulated conductivity problem with pp-Laplacian

We study the insulated conductivity problem with closely spaced insulators embedded in a homogeneous matrix where the current-electric field relation is the power law $J = |E|^{p-2}E$. The gradient of solutions may blow up as $\varepsilon$, the distance between insulators, approaches to 0. In 2D, we prove an upper bound of the gradient to be of order $\varepsilon^{-\alpha}$, where $\alpha = 1/2$ when $p \in(1,3]$ and any $\alpha > 1/(p-1)$ when $p > 3$. We provide examples to show that this exponent is almost optimal. In dimensions $n \ge 3$, we prove an upper bound of order $\varepsilon^{-1/2 + \beta}$ for some $\beta > 0$, and show that $\beta \nearrow 1/2$ as $n \to \infty$.Comment: 39 pages. Theorem 1.3 is extended to all dimension

Peak wind pressures on roof claddings of regular railway stations

Small or medium-scales regular railway stations are being widely built. Through an extensive survey of actual modern railway stations, the common ranges of building parameters were determined. Wind pressure measurements of 18 models were conducted by wind tunnel tests to systematically investigate the influence of the canopy width, opening width and waiting hall building height on peak wind pressure coefficients on roof claddings of the canopies and waiting hall building. According to the distribution law of the most critical minimum wind pressure coefficients, roofs of the two canopies and waiting hall building are divided into 19 zones for the convenience of engineering applications. The dependency of the area-averaged most critical minimum wind pressure coefficients on the tributary area in all roof zones of the canopies and waiting hall building are investigated. The fitted formulas to calculate the area-averaged most critical wind pressure coefficients are proposed as a function of the tributary area, which can be used as the reference for determining wind loads during the design of roof claddings of regular railway stations

AutoGraph: Optimizing DNN Computation Graph for Parallel GPU Kernel Execution

Deep learning frameworks optimize the computation graphs and intra-operator computations to boost the inference performance on GPUs, while inter-operator parallelism is usually ignored. In this paper, a unified framework, AutoGraph, is proposed to obtain highly optimized computation graphs in favor of parallel executions of GPU kernels. A novel dynamic programming algorithm, combined with backtracking search, is adopted to explore the optimal graph optimization solution, with the fast performance estimation from the mixed critical path cost. Accurate runtime information based on GPU Multi-Stream launched with CUDA Graph is utilized to determine the convergence of the optimization. Experimental results demonstrate that our method achieves up to 3.47x speedup over existing graph optimization methods. Moreover, AutoGraph outperforms state-of-the-art parallel kernel launch frameworks by up to 1.26x