Admissible function spaces for weighted Sobolev inequalities

Let $k,N \in \mathbb{N}$ with $1\le k\le N$ and let $\Omega=\Omega_1 \times \Omega_2$ be an open set in $\mathbb{R}^k \times \mathbb{R}^{N-k}$. For $p\in (1,\infty)$ and $q \in (0,\infty),$ we consider the following Hardy-Sobolev type inequality: \begin{align} \int_{\Omega} |g_1(y)g_2(z)| |u(y,z)|^q \, dy \, dz \leq C \left( \int_{\Omega} | \nabla u(y,z) |^p \, dy \, dz \right)^{\frac{q}{p}}, \quad \forall \, u \in \mathcal{C}^1_c(\Omega), \end{align} for some $C>0$. Depending on the values of $N,k,p,q,$ we have identified various pairs of Lorentz spaces, Lorentz-Zygmund spaces and weighted Lebesgue spaces for $(g_1, g_2)$ so that the above inequality holds. Furthermore, we give a sufficient condition on $g_1,g_2$ so that the best constant in the above inequality is attained in the Beppo-Levi space $\mathcal{D}^{1,p}_0(\omega)$-the completion of $\mathcal{C}^1_c(\Omega)$ with respect to $\|\nabla u\|_{L^p(\Omega)}$.Comment: 40 page

On the generalized weighted Sobolev inequality

Let $\Omega$ be an open subset of $\mathbb{R}^N.$ We identify various classes of Young functions $\Phi,\,\Psi$, and weight functions $g\in L^1_\text{loc}(\Omega)$ so that the following generalized weighted Sobolev inequality holds: \begin{equation*}\label{ineq:Orlicz} \Psi^{-1}\left(\int_{\Omega}|g(x)|\Psi( |u(x)| )dx \right)\leq C\Phi^{-1}\left(\int_{\Omega}\Phi(|\nabla u(x)|) dx \right),\,\,\,\forall\,u\in \mathcal{C}^1_c(\Omega), \end{equation*} for some $C>0$. As an application, we study the existence of non-negative solutions for certain nonlinear weighted eigenvalue problems.Comment: 27 page

Impact of noncardiac findings in patients undergoing CT coronary angiography:a substudy of the Scottish computed tomography of the heart (SCOT-HEART) trial

Objectives Noncardiac findings are common on coronary computed tomography angiography (CCTA). We assessed the clinical impact of noncardiac findings, and potential changes to surveillance scans with the application of new lung nodule guidelines. Methods This substudy of the SCOT-HEART randomized controlled trial assessed noncardiac findings identified on CCTA. Clinically significant noncardiac findings were those causing symptoms or requiring further investigation, follow-up or treatment. Lung nodule follow-up was undertaken following the 2005 Fleischner guidelines. The potential impact of the 2015 British Thoracic Society (BTS) and the 2017 Fleischner guidelines was assessed. Results CCTA was performed in 1,778 patients and noncardiac findings were identified in 677 (38%). In 173 patients (10%) the abnormal findings were clinically significant and in 55 patients (3%) the findings were the cause of symptoms. Follow-up imaging was recommended in 136 patients (7.6%) and additional clinic consultations were organized in 46 patients (2.6%). Malignancy was diagnosed in 7 patients (0.4%). Application of the new lung nodule guidelines would have reduced the number of patients undergoing a follow-up CT scan: 68 fewer with the 2015 BTS guidelines and 78 fewer with the 2017 Fleischner guidelines; none of these patients subsequently developed malignancy. Conclusions Clinically significant noncardiac findings are identified in 10% of patients undergoing CCTA. Application of new lung nodule guidelines will reduce the cost of surveillance, without the risk of missing malignancy

Draft genome sequence of Sclerospora graminicola, the pearl millet downy mildew pathogen:Genome sequence of pearl millet downy mildew pathogen

Sclerospora graminicola pathogen is one of the most important biotic production constraints of pearl millet worldwide. We report a de novo whole genome assembly and analysis of pathotype 1. The draft genome assembly contained 299,901,251 bp with 65,404 genes. Pearl millet [Pennisetum glaucum (L.) R. Br.], is an important crop of the semi-arid and arid regions of the world. It is capable of growing in harsh and marginal environments with highest degree of tolerance to drought and heat among cereals (1). Downy mildew is the most devastating disease of pearl millet caused by Sclerospora graminicola (sacc. Schroet), particularly on genetically uniform hybrids. Estimated annual grain yield loss due to downy mildew is approximately 10?80 % (2-7). Pathotype 1 has been reported to be the highly virulent pathotype of Sclerospora graminicola in India (8). We report a de novo whole genome assembly and analysis of Sclerospora graminicola pathotype 1 from India. A susceptible pearl millet genotype Tift 23D2B1P1-P5 was used for obtaining single-zoospore isolates from the original oosporic sample. The library for whole genome sequencing was prepared according to the instructions by NEB ultra DNA library kit for Illumina (New England Biolabs, USA). The libraries were normalised, pooled and sequenced on Illumina HiSeq 2500 (Illumina Inc., San Diego, CA, USA) platform at 2 x100 bp length. Mate pair (MP) libraries were prepared using the Nextera mate pair library preparation kit (Illumina Inc., USA). 1 ?g of Genomic DNA was subject to tagmentation and was followed by strand displacement. Size selection tagmented/strand displaced DNA was carried out using AmpureXP beads. The libraries were validated using an Agilent Bioanalyser using DNA HS chip. The libraries were normalised, pooled and sequenced on Illumina MiSeq (Illumina Inc., USA) platform at 2 x300 bp length. The whole genome sequencing was performed by sequencing of 7.38 Gb with 73,889,924 paired end reads from paired end library, and 1.15 Gb with 3,851,788 reads from mate pair library generated from Illumina HiSeq2500 and Illumina MiSeq, respectively. The sequences were assembled using various assemblers like ABySS, MaSuRCA, Velvet, SOAPdenovo2, and ALLPATHS-LG. The assembly generated by MaSuRCA (9) algorithm was observed superior over other algorithms and hence used for scaffolding using SSPACE. Assembled draft genome sequence of S. graminicola pathotype 1 was 299,901,251 bp long, with a 47.2 % GC content consisting of 26,786 scaffolds with N50 of 17,909 bp with longest scaffold size of 238,843 bp. The overall coverage was 40X. The draft genome sequence was used for gene prediction using AUGUSTUS. The completeness of the assembly was investigated using CEGMA and revealed 92.74% proteins completely present and 95.56% proteins partially present, while BUSCO fungal dataset indicated 64.9% complete, 12.4% fragmented, 22.7% missing out of 290 BUSCO groups. A total of 52,285 predicted genes were annotated using BLASTX and 38,120 genes were observed with significant BLASTX match. Repetitive element analysis in the assembly revealed 8,196 simple repeats, 1,058 low complexity repeats and 5,562 dinucleotide to hexanucleotide microsatellite repeats.publishersversionPeer reviewe