Existence of Integral mm-Varifolds minimizing ∫∣A∣p\int |A|^p and ∫∣H∣p\int |H|^p, p>mp>m, in Riemannian Manifolds

We prove existence and partial regularity of integral rectifiable $m$-dimensional varifolds minimizing functionals of the type $\int |H|^p$ and $\int |A|^p$ in a given Riemannian $n$-dimensional manifold $(N,g)$, $2\leq mm$, under suitable assumptions on $N$ (in the end of the paper we give many examples of such ambient manifolds). To this aim we introduce the following new tools: some monotonicity formulas for varifolds in $\mathbb{R}^S$ involving $\int |H|^p$, to avoid degeneracy of the minimizer, and a sort of isoperimetric inequality to bound the mass in terms of the mentioned functionals.Comment: 33 pages; this second submission corresponds to the published version of the paper, minor typos are fixe

Physics in Riemann's mathematical papers

Riemann's mathematical papers contain many ideas that arise from physics, and some of them are motivated by problems from physics. In fact, it is not easy to separate Riemann's ideas in mathematics from those in physics. Furthermore, Riemann's philosophical ideas are often in the background of his work on science. The aim of this chapter is to give an overview of Riemann's mathematical results based on physical reasoning or motivated by physics. We also elaborate on the relation with philosophy. While we discuss some of Riemann's philosophical points of view, we review some ideas on the same subjects emitted by Riemann's predecessors, and in particular Greek philosophers, mainly the pre-socratics and Aristotle. The final version of this paper will appear in the book: From Riemann to differential geometry and relativity (L. Ji, A. Papadopoulos and S. Yamada, ed.) Berlin: Springer, 2017

Index estimates for free boundary minimal hypersurfaces

We show that the Morse index of a properly embedded free boundary minimal hypersurface in a strictly mean convex domain of the Euclidean space grows linearly with the dimension of its first relative homology group (which is at least as big as the number of its boundary components, minus one). In ambient dimension three, this implies a lower bound for the index of a free boundary minimal surface which is linear both with respect to the genus and the number of boundary components. Thereby, the compactness theorem by Fraser and Li implies a strong compactness theorem for the space of free boundary minimal surfaces with uniformly bounded Morse index inside a convex domain. Our estimates also imply that the examples constructed, in the unit ball, by Fraser–Schoen and Folha–Pacard–Zolotareva have arbitrarily large index. Extensions of our results to more general settings (including various classes of positively curved Riemannian manifolds and other convexity assumptions) are discussed

Population specificity of the DNAI1 gene mutation spectrum in primary ciliary dyskinesia (PCD)

<p>Abstract</p> <p>Background</p> <p>Mutations in the <it>DNAI1 </it>gene, encoding a component of outer dynein arms of the ciliary apparatus, are the second most important genetic cause of primary ciliary dyskinesia (PCD), the genetically heterogeneous recessive disorder with the prevalence of ~1/20,000. The estimates of the <it>DNAI1 </it>involvement in PCD pathogenesis differ among the reported studies, ranging from 4% to 10%.</p> <p>Methods</p> <p>The coding sequence of <it>DNAI1 </it>was screened (SSCP analysis and direct sequencing) in a group of PCD patients (157 families, 185 affected individuals), the first ever studied large cohort of PCD patients of Slavic origin (mostly Polish); multiplex ligation-dependent probe amplification (MLPA) analysis was performed in a subset of ~80 families.</p> <p>Results</p> <p>Three previously reported mutations (IVS1+2-3insT, L513P and A538T) and two novel missense substitutions (C388Y and G515S) were identified in 12 families (i.e. ~8% of non-related Polish PCD patients). The structure of background SNP haplotypes indicated common origin of each of the two most frequent mutations, IVS1+2-3insT and A538T. MLPA analysis did not reveal any significant differences between patients and control samples. The Polish cohort was compared with all the previously studied PCD groups (a total of 487 families): IVS1+2-3insT remained the most prevalent pathogenetic change in <it>DNAI1 </it>(54% of the mutations identified worldwide), and the increased global prevalence of A538T (14%) was due to the contribution of the Polish cohort.</p> <p>Conclusions</p> <p>The worldwide involvement of <it>DNAI1 </it>mutations in PCD pathogenesis in families not preselected for ODA defects ranges from 7 to 10%; this global estimate as well as the mutation profile differs in specific populations. Analysis of the background SNP haplotypes suggests that the increased frequency of chromosomes carrying A538T mutations in Polish patients may reflects local (Polish or Slavic) founder effect. Results of the MLPA analysis indicate that no large exonic deletions are involved in PCD pathogenesis.</p

Magnesium induces neuronal apoptosis by suppressing excitability

In clinical obstetrics, magnesium sulfate (MgSO4) use is widespread, but effects on brain development are unknown. Many agents that depress neuronal excitability increase developmental neuroapoptosis. In this study, we used dissociated cultures of rodent hippocampus to examine the effects of Mg++ on excitability and survival. Mg++-induced caspase-3-associated cell loss at clinically relevant concentrations. Whole-cell patch-clamp techniques measured Mg++ effects on action potential threshold, action potential peak amplitude, spike number and changes in resting membrane potential. Mg++ depolarized action potential threshold, presumably from surface charge screening effects on voltage-gated sodium channels. Mg++ also decreased the number of action potentials in response to fixed current injection without affecting action potential peak amplitude. Surprisingly, Mg++ also depolarized neuronal resting potential in a concentration-dependent manner with a +5.2 mV shift at 10 mM. Voltage ramps suggested that Mg++ blocked a potassium conductance contributing to the resting potential. In spite of this depolarizing effect of Mg++, the net inhibitory effect of Mg++ nearly completely silenced neuronal network activity measured with multielectrode array recordings. We conclude that although Mg++ has complex effects on cellular excitability, the overall inhibitory influence of Mg++ decreases neuronal survival. Taken together with recent in vivo evidence, our results suggest that caution may be warranted in the use of Mg++ in clinical obstetrics and neonatology