Optimal and pressure-independent L2 velocity error estimates for a modified Crouzeix-Raviart Stokes element with BDM reconstructions

Nearly all inf-sup stable mixed finite elements for the incompressible Stokes equations relax the divergence constraint. The price to pay is that a priori estimates for the velocity error become pressure-dependent, while divergence-free mixed finite elements deliver pressure-independent estimates. A recently introduced new variational crime using lowest-order Raviart-Thomas velocity reconstructions delivers a much more robust modified Crouzeix-Raviart element, obeying an optimal pressure-independent discrete H1 velocity estimate. Refining this approach, a more sophisticated variational crime employing the lowest-order BDM element is proposed, which also allows proving an optimal pressure-independent L2 velocity error. Numerical examples confirm the analysis and demonstrate the improved robustness in the Navier-Stokes case

Operator Norm Bounds on the Correlation Matrix of the SK Model at High Temperature

We prove that the two point correlation matrix $\textbf{M}= (\langle \sigma_i ; \sigma_j\rangle)_{1\leq i,j\leq N} \in \mathbb{R}^{N\times N}$ of the Sherrington-Kirkpatrick model has the property that for every $\epsilon>0$ there exists $K_\epsilon>0$, that is independent of $N$, such that $$\mathbb{P}\big( \| \textbf{M} \|_{\text{op}} \leq K_{\epsilon}\big) \geq 1- \epsilon$$ for $N$ large enough, for suitable interaction and external field parameters $(\beta,h)$ in the replica symmetric region. In other words, the operator norm of $\textbf{M}$ is of order one with high probability. Our results are in particular valid for all $(\beta,h)\in (0,1)\times (0,\infty)$ and thus complement recently obtained results in \cite{EAG,BSXY} that imply the operator norm boundedness of $\textbf{M}$ for all $\beta<1$ in the special case of vanishing external field.Comment: 33 page

Optimal Rate for Bose-Einstein Condensation in the Gross-Pitaevskii Regime

We consider systems of bosons trapped in a box, in the Gross-Pitaevskii regime. We show that low-energy states exhibit complete Bose-Einstein condensation with an optimal bound on the number of orthogonal excitations. This extends recent results obtained in \cite{BBCS1}, removing the assumption of small interaction potential.Comment: 99 pages, typos correcte

Gene expression profiling in whole blood of patients with coronary artery disease

Owing to the dynamic nature of the transcriptome, gene expression profiling is a promising tool for discovery of disease-related genes and biological pathways. In the present study, we examined gene expression in whole blood of 12 patients with CAD (coronary artery disease) and 12 healthy control subjects. Furthermore, ten patients with CAD underwent whole-blood gene expression analysis before and after the completion of a cardiac rehabilitation programme following surgical coronary revascularization. mRNA and miRNA (microRNA) were isolated for expression profiling. Gene expression analysis identified 365 differentially expressed genes in patients with CAD compared with healthy controls (175 up- and 190 down-regulated in CAD), and 645 in CAD rehabilitation patients (196 up- and 449 down-regulated post-rehabilitation). Biological pathway analysis identified a number of canonical pathways, including oxidative phosphorylation and mitochondrial function, as being significantly and consistently modulated across the groups. Analysis of miRNA expression revealed a number of differentially expressed miRNAs, including hsa-miR-140-3p (control compared with CAD, P=0.017), hsa-miR-182 (control compared with CAD, P=0.093), hsa-miR-92a and hsa-miR-92b (post- compared with pre-exercise, P<0.01). Global analysis of predicted miRNA targets found significantly reduced expression of genes with target regions compared with those without: hsa-miR-140-3p (P=0.002), hsa-miR-182 (P=0.001), hsa-miR-92a and hsa-miR-92b (P=2.2×10−16). In conclusion, using whole blood as a ‘surrogate tissue’ in patients with CAD, we have identified differentially expressed miRNAs, differentially regulated genes and modulated pathways which warrant further investigation in the setting of cardiovascular function. This approach may represent a novel non-invasive strategy to unravel potentially modifiable pathways and possible therapeutic targets in cardiovascular disease