A general wavelet-based profile decomposition in the critical embedding of function spaces

We characterize the lack of compactness in the critical embedding of functions spaces $X\subset Y$ having similar scaling properties in the following terms : a sequence $(u_n)_{n\geq 0}$ bounded in $X$ has a subsequence that can be expressed as a finite sum of translations and dilations of functions $(\phi_l)_{l>0}$ such that the remainder converges to zero in $Y$ as the number of functions in the sum and $n$ tend to $+\infty$. Such a decomposition was established by G\'erard for the embedding of the homogeneous Sobolev space $X=\dot H^s$ into the $Y=L^p$ in $d$ dimensions with $0<s=d/2-d/p$, and then generalized by Jaffard to the case where $X$ is a Riesz potential space, using wavelet expansions. In this paper, we revisit the wavelet-based profile decomposition, in order to treat a larger range of examples of critical embedding in a hopefully simplified way. In particular we identify two generic properties on the spaces $X$ and $Y$ that are of key use in building the profile decomposition. These properties may then easily be checked for typical choices of $X$ and $Y$ satisfying critical embedding properties. These includes Sobolev, Besov, Triebel-Lizorkin, Lorentz, H\"older and BMO spaces.Comment: 24 page

Concentration analysis and cocompactness

Loss of compactness that occurs in may significant PDE settings can be expressed in a well-structured form of profile decomposition for sequences. Profile decompositions are formulated in relation to a triplet $(X,Y,D)$, where $X$ and $Y$ are Banach spaces, $X\hookrightarrow Y$, and $D$ is, typically, a set of surjective isometries on both $X$ and $Y$. A profile decomposition is a representation of a bounded sequence in $X$ as a sum of elementary concentrations of the form $g_kw$, $g_k\in D$, $w\in X$, and a remainder that vanishes in $Y$. A necessary requirement for $Y$ is, therefore, that any sequence in $X$ that develops no $D$-concentrations has a subsequence convergent in the norm of $Y$. An imbedding $X\hookrightarrow Y$ with this property is called $D$-cocompact, a property weaker than, but related to, compactness. We survey known cocompact imbeddings and their role in profile decompositions

Studying complex interventions : reflections from the FEMHealth project on evaluating fee exemption policies in West Africa and Morocco

Peer reviewedPublisher PD

Ensemble approach for generalized network dismantling

Finding a set of nodes in a network, whose removal fragments the network below some target size at minimal cost is called network dismantling problem and it belongs to the NP-hard computational class. In this paper, we explore the (generalized) network dismantling problem by exploring the spectral approximation with the variant of the power-iteration method. In particular, we explore the network dismantling solution landscape by creating the ensemble of possible solutions from different initial conditions and a different number of iterations of the spectral approximation.Comment: 11 Pages, 4 Figures, 4 Table

Urinary Exosomal microRNA-451-5p Is a Potential Early Biomarker of Diabetic Nephropathy in Rats

Non-invasive renal signatures can help in serial monitoring of diabetic patients. We tested whether urinary exosomal (UE) microRNA (miR) analysis could non-invasively predict renal pathology in diabetic rats during the course of diabetes. Diabetes mellitus (DM) was induced in male Wistar rats by a single intraperitoneal injection of streptozotocin (STZ, 50 mg/kg body weight). Non-diabetic control (CTRL) rats were injected with vehicle. Insulin (INS) treatment (5U/d, s.c.) was provided to 50% of the DM rats. Urine samples were collected at weeks 3, 6, and 9 following injections and UE prepared. An increase in miR-451-5p and miR-16, observed by pilot small RNA sequencing of UE RNA, was confirmed by quantitative real-time polymerase chain reaction (qPCR) and selected for further study. Subsets of rats were euthanized after 3, 6, and 9 weeks of diabetes for renal pathology analysis, including determination of the tubulointerstitial fibrotic index (TFI) and glomerulosclerotic index (GI) scores. qPCR showed a substantial rise in miR-451-5p in UE from DM rats during thecourse of diabetes, with a significant rise (median fold change >1000) between 3 and 6 weeks. Moreover, UE miR-451-5p at 6 weeks predicted urine albumin at 9 weeks (r = 0.76). A delayed but significant rise was also observed for miR-16. In contrast, mean urine albumin only increased 21% between 3 and 6 weeks (non-significant rise), and renal TFI and GI were unchanged till 9 weeks. Renal expression of miR-451-5p and miR-16 (at 10 weeks) did not correlate with urine levels, and moreover, was negatively associated with indices of renal pathology (r�-0.70, p = 0.005 for TFI and r�-0.6, p�0.02 for GI). Overall, a relative elevation in renal miR-451-5p and miR-16 in diabetes appeared protective against diabetes- induced kidney fibrosis; while UE miR-451-5p may hold prognostic value as an earlyand sensitive non-invasive indicator of renal diseas

Sequencing of high-complexity DNA pools for identification of nucleotide and structural variants in regions associated with complex traits

We have used targeted genomic sequencing of high-complexity DNA pools based on long-range PCR and deep DNA sequencing by the SOLiD technology. The method was used for sequencing of 286 kb from four chromosomal regions with quantitative trait loci (QTL) influencing blood plasma lipid and uric acid levels in DNA pools of 500 individuals from each of five European populations. The method shows very good precision in estimating allele frequencies as compared with individual genotyping of SNPs (r(2) = 0.95, P &lt; 10(-16)). Validation shows that the method is able to identify novel SNPs and estimate their frequency in high-complexity DNA pools. In our five populations, 17% of all SNPs and 61% of structural variants are not available in the public databases. A large fraction of the novel variants show a limited geographic distribution, with 62% of the novel SNPs and 59% of novel structural variants being detected in only one of the populations. The large number of population-specific novel SNPs underscores the need for comprehensive sequencing of local populations in order to identify the causal variants of human traits