The critical dimension for a 4th order problem with singular nonlinearity

We study the regularity of the extremal solution of the semilinear biharmonic equation \bi u=\f{\lambda}{(1-u)^2}, which models a simple Micro-Electromechanical System (MEMS) device on a ball B\subset\IR^N, under Dirichlet boundary conditions $u=\partial_\nu u=0$ on $\partial B$. We complete here the results of F.H. Lin and Y.S. Yang \cite{LY} regarding the identification of a "pull-in voltage" \la^*>0 such that a stable classical solution u_\la with 0 exists for \la\in (0,\la^*), while there is none of any kind when \la>\la^*. Our main result asserts that the extremal solution $u_{\lambda^*}$ is regular $(\sup_B u_{\lambda^*} <1)$ provided $N \le 8$ while $u_{\lambda^*}$ is singular ($\sup_B u_{\lambda^*} =1$) for $N \ge 9$, in which case $1-C_0|x|^{4/3}\leq u_{\lambda^*} (x) \leq 1-|x|^{4/3}$ on the unit ball, where $C_0:= (\frac{\lambda^*}{\overline{\lambda}})^{1/3}$ and $\bar{\lambda}:= {8/9} (N-{2/3}) (N- {8/3})$.Comment: 19 pages. This paper completes and replaces a paper (with a similar title) which appeared in arXiv:0810.5380. Updated versions --if any-- of this author's papers can be downloaded at this http://www.birs.ca/~nassif

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

Social distance during the covid-19 pandemic reflects perceived rather than actual risk

Interpersonal space (IPS) is the area surrounding our own bodies in which we interact comfortably with other individuals. During the COVID-19 pandemic, keeping larger IPS than usual, along with wearing a face mask, is one of the most effective measures to slow down the COVID-19 outbreak. Here, we explore the contribution of actual and perceived risk of contagion and anxiety levels in regulating our preferred social distance from other people during the first wave of the COVID-19 pandemic in Italy. In this study, 1293 individuals from six Italian regions with different levels of actual risk of infection participated in an online survey assessing their perceived risk to be infected, level of anxiety and IPS. Two tasks were adopted as measures of interpersonal distance: the Interpersonal Visual Analogue Scale and a questionnaire evaluating interpersonal distance with and without face mask. The results showed that the IPS regulation was affected by how people subjectively perceived COVID-19 risk and the related level of anxiety, not by actual objective risk. This clarifies that the role of threat in prompting avoidant behaviors expressed in increased IPS does not merely reflect environmental events but rather how they are subjectively experienced and represented

Natural preconditioning and iterative methods for saddle point systems

The solution of quadratic or locally quadratic extremum problems subject to linear(ized) constraints gives rise to linear systems in saddle point form. This is true whether in the continuous or the discrete setting, so saddle point systems arising from the discretization of partial differential equation problems, such as those describing electromagnetic problems or incompressible flow, lead to equations with this structure, as do, for example, interior point methods and the sequential quadratic programming approach to nonlinear optimization. This survey concerns iterative solution methods for these problems and, in particular, shows how the problem formulation leads to natural preconditioners which guarantee a fast rate of convergence of the relevant iterative methods. These preconditioners are related to the original extremum problem and their effectiveness---in terms of rapidity of convergence---is established here via a proof of general bounds on the eigenvalues of the preconditioned saddle point matrix on which iteration convergence depends

Il percorso ideale nella gestione del paziente con osteoporosi severa:dalla valutazione del rischio all'approccio multidisciplinare

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GeneCAT—novel webtools that combine BLAST and co-expression analyses

The gene co-expression analysis toolbox (GeneCAT) introduces several novel microarray data analyzing tools. First, the multigene co-expression analysis, combined with co-expressed gene networks, provides a more powerful data mining technique than standard, single-gene co-expression analysis. Second, the high-throughput Map-O-Matic tool matches co-expression pattern of multiple query genes to genes present in user-defined subdatabases, and can therefore be used for gene mapping in forward genetic screens. Third, Rosetta combines co-expression analysis with BLAST and can be used to find ‘true’ gene orthologs in the plant model organisms Arabidopsis thaliana and Hordeum vulgare (Barley). GeneCAT is equipped with expression data for the model plant A. thaliana, and first to introduce co-expression mining tools for the monocot Barley. GeneCAT is available at http://genecat.mpg.d

Lactobacillus helveticus MIMLh5-Specific Antibodies for Detection of S-Layer Protein in Grana Padano Protected-Designation-of-Origin Cheese

Single-chain variable-fragment antibodies (scFvs) have considerable potential in immunological detection and localization of bacterial surface structures. In this study, synthetic phage-displayed antibody libraries were used to select scFvs against immunologically active S-layer protein of Lactobacillus helveticus MIMLh5. After three rounds of panning, five relevant phage clones were obtained, of which four were specific for the S-layer protein of L. helveticus MIMLh5 and one was also capable of binding to the S-layer protein of L. helveticus ATCC 15009. All five anti-S-layer scFvs were expressed in Escherichia coli XL1-Blue, and their specificity profiles were characterized by Western blotting. The anti-S-layer scFv PolyH4, with the highest specificity for the Slayer protein of L. helveticus MIMLh5, was used to detect the S-layer protein in Grana Padano protected-designation-of-origin (PDO) cheese extracts by Western blotting. These results showed promising applications of this monoclonal antibody for the detection of immunomodulatory S-layer protein in dairy (and dairy-based) foods

Alkalizing Reactions Streamline Cellular Metabolism in Acidogenic Microorganisms

An understanding of the integrated relationships among the principal cellular functions that govern the bioenergetic reactions of an organism is necessary to determine how cells remain viable and optimise their fitness in the environment. Urease is a complex enzyme that catalyzes the hydrolysis of urea to ammonia and carbonic acid. While the induction of urease activity by several microorganisms has been predominantly considered a stress-response that is initiated to generate a nitrogen source in response to a low environmental pH, here we demonstrate a new role of urease in the optimisation of cellular bioenergetics. We show that urea hydrolysis increases the catabolic efficiency of Streptococcus thermophilus, a lactic acid bacterium that is widely used in the industrial manufacture of dairy products. By modulating the intracellular pH and thereby increasing the activity of β-galactosidase, glycolytic enzymes and lactate dehydrogenase, urease increases the overall change in enthalpy generated by the bioenergetic reactions. A cooperative altruistic behaviour of urease-positive microorganisms on the urease-negative microorganisms within the same environment was also observed. The physiological role of a single enzymatic activity demonstrates a novel and unexpected view of the non-transcriptional regulatory mechanisms that govern the bioenergetics of a bacterial cell, highlighting a new role for cytosol-alkalizing biochemical pathways in acidogenic microorganisms