In the article, we present several quadratic transformation inequalities for Gaussian hypergeometric function and find the analogs of duplication inequalities for the generalized Grötzsch ring function.

Transcriptome-wide association studies integrate gene expression data with common risk variation to identify gene-trait associations. By incorporating epigenome data to estimate the functional importance of genetic variation on gene expression, we generate a small but significant improvement in the accuracy of transcriptome prediction and increase the power to detect significant expression-trait associations. Joint analysis of 14 large-scale transcriptome datasets and 58 traits identify 13,724 significant expression-trait associations that converge on biological processes and relevant phenotypes in human and mouse phenotype databases. We perform drug repurposing analysis and identify compounds that mimic, or reverse, trait-specific changes. We identify genes that exhibit agonistic pleiotropy for genetically correlated traits that converge on shared biological pathways and elucidate distinct processes in disease etiopathogenesis. Overall, this comprehensive analysis provides insight into the specificity and convergence of gene expression on susceptibility to complex traits.

In the article, we prove that the double inequality x 2 + p 0 x + p 0 < Γ ( x + 1 ) < x 2 + 9 / 5 x + 9 / 5 $$ \frac{x^{2}+p_{0}}{x+p_{0}}< \Gamma(x+1)< \frac{x^{2}+9/5}{x+9/5} $$ holds for all x ∈ ( 0 , 1 ) $x\in(0, 1)$ , we present the best possible constants λ and μ such that λ ( x 2 + 9 / 5 ) x + 9 / 5 ≤ Γ ( x + 1 ) ≤ μ ( x 2 + p 0 ) x + p 0 $$ \frac{\lambda(x^{2}+9/5)}{x+9/5}\leq\Gamma(x+1)\leq\frac{\mu (x^{2}+p_{0})}{x+p_{0}} $$ for all x ∈ ( 0 , 1 ) $x\in(0, 1)$ , and we find the value of x ∗ $x^{\ast}$ in the interval ( 0 , 1 ) $(0, 1)$ such that Γ ( x + 1 ) > ( x 2 + 1 / γ ) / ( x + 1 / γ ) $\Gamma(x+1)>(x^{2}+1/\gamma)/(x+1/\gamma)$ for x ∈ ( 0 , x ∗ ) $x\in(0, x^{\ast})$ and Γ ( x + 1 ) < ( x 2 + 1 / γ ) / ( x + 1 / γ ) $\Gamma(x+1)<(x^{2}+1/\gamma)/(x+1/\gamma )$ for x ∈ ( x ∗ , 1 ) $x\in(x^{\ast}, 1)$ , where Γ ( x ) $\Gamma(x)$ is the classical gamma function, γ = lim n → ∞ ( ∑ k = 1 n 1 / k − log n ) = 0.577 … $\gamma=\lim_{n\rightarrow\infty}(\sum_{k=1}^{n}1/k-\log n)=0.577\ldots$ is Euler-Mascheroni constant and p 0 = γ / ( 1 − γ ) = 1.365 … $p_{0}=\gamma/(1-\gamma )=1.365\ldots$ .

Background: The 2019 novel coronavirus (COVID-19) has continuous outbreaks around the world. Lung is the main organ that be involved. There is a lack of clinical data on the respiratory sounds of COVID-19 infected pneumonia, which includes invaluable information concerning physiology and pathology. The medical resources are insufficient, which are now mainly supplied for the severe patients. The development of a convenient and effective screening method for mild or asymptomatic suspicious patients is highly demanded. Methods: This is a retrospective case series study. 10 patients with positive results of nucleic acid were enrolled in this study. Lung auscultation was performed by the same physician on admission using a hand-held portable electronic stethoscope delivered in real time via Bluetooth. The recorded audio was exported, and was analyzed by six physicians. Each physician individually described the abnormal breathing sounds that he heard. The results were analyzed in combination with clinical data. Signal analysis was used to quantitatively describe the most common abnormal respiratory sounds. Results: All patients were found abnormal breath sounds at least by 3 physicians, and one patient by all physicians. Cackles, asymmetrical vocal resonance and indistinguishable murmurs are the most common abnormal breath sounds. One asymptomatic patient was found vocal resonance, and the result was correspondence with radiographic computed tomography. Signal analysis verified the credibility of the above abnormal breath sounds. Conclusions: This study describes respiratory sounds of patients with COVID-19, which fills up for the lack of clinical data and provides a simple screening method for suspected patients.