Newtonian Lorentz Metric Spaces

This paper studies Newtonian Sobolev-Lorentz spaces. We prove that these spaces are Banach. We also study the global p,q-capacity and the p,q-modulus of families of rectifiable curves. Under some additional assumptions (that is, the space carries a doubling measure and a weak Poincare inequality) and some restrictions on q, we show that the Lipschitz functions are dense in those spaces. Moreover, in the same setting we show that the p,q-capacity is Choquet provided that q is strictly greater than 1. We also provide a counterexample to the density result of Lipschitz functions in the Euclidean setting when q is infinite.Comment: v2: 32 pages. Formula on page 23 corrected; typos remove

Boundary measures, generalized Gauss-Green formulas, and mean value property in metric measure spaces

We study mean value properties of harmonic functions in metric measure spaces. The metric measure spaces we consider have a doubling measure and support a (1,1)- Poincar\'e inequality. The notion of harmonicity is based on the Dirichlet form defined in terms of a Cheeger differentiable structure. By studying fine properties of the Green function on balls, we characterize harmonic functions in terms of a mean value property. As a consequence, we obtain a detailed description of Poisson kernels. We shall also obtain a Gauss-Green type formula for sets of finite perimeter which posses a Minkowski content characterization of the perimeter. For the Gauss-Green formula we introduce a suitable notion of the interior normal trace of a regular ball

Two characterization of BV functions on Carnot groups via the heat semigroup

In this paper we provide two different characterizations of sets with finite perimeter and functions of bounded variation in Carnot groups, analogous to those which hold in Euclidean spaces, in terms of the short-time behaviour of the heat semigroup. The second one holds under the hypothesis that the reduced boundary of a set of finite perimeter is rectifiable, a result that presently is known in Step 2 Carnot groups

Characterizations of Sobolev spaces on sublevel sets in abstract Wiener spaces

In this paper we consider an abstract Wiener space $(X,\gamma,H)$ and an open subset $O\subseteq X$ which satisfies suitable assumptions. For every $p\in(1,+\infty)$ we define the Sobolev space $W_{0}^{1,p}(O,\gamma)$ as the closure of Lipschitz continuous functions which support with positive distance from $\partial O$ with respect to the natural Sobolev norm, and we show that under the assumptions on $O$ the space $W_{0}^{1,p}(O,\gamma)$ can be characterized as the space of functions in $W^{1,p}(O,\gamma)$ which have null trace at the boundary $\partial O$, or, equivalently, as the space of functions defined on $O$ whose trivial extension belongs to $W^{1,p}(X,\gamma)$

Some isoperimetric problems in planes with density

We study the isoperimetric problem in Euclidean space endowed with a density. We first consider piecewise constant densities and examine particular cases related to the characteristic functions of half-planes, strips and balls. We also consider continuous modification of Gauss density in $\R^2$. Finally, we give a list of related open questions.Comment: 40 pages, 19 figure

On BVBV functions and essentially bounded divergence-measure fields in metric spaces

By employing the differential structure recently developed by N. Gigli, we first give a notion of functions of bounded variation ($BV$) in terms of suitable vector fields on a complete and separable metric measure space $(\mathbb{X},d,\mu)$ equipped with a non-negative Radon measure $\mu$ finite on bounded sets. Then, we extend the concept of divergence-measure vector fields $\mathcal{DM}^p(\mathbb{X})$ for any $p\in[1,\infty]$ and, by simply requiring in addition that the metric space is locally compact, we determine an appropriate class of domains for which it is possible to obtain a Gauss-Green formula in terms of the normal trace of a $\mathcal{DM}^\infty(\mathbb{X})$ vector field. This differential machinery is also the natural framework to specialize our analysis for ${\mathsf{RCD}(K,\infty)}$ spaces, where we exploit the underlying geometry to determine the Leibniz rules for $\mathcal{DM}^\infty(\mathbb{X})$ and ultimately to extend our discussion on the Gauss-Green formulas.Comment: 64 pages, 0 figures; accepted, to appear in Rev. Mat. Iberoamerican

Some Fine Properties of BV Functions on Wiener Spaces

Abstract In this paper we define jump set and approximate limits for BV functions on Wiener spaces and show that the weak gradient admits a decomposition similar to the finite dimensional case. We also define the SBV class of functions of special bounded variation and give a characterisation of SBV via a chain rule and a closure theorem. We also provide a characterisation of BV functions in terms of the short-time behaviour of the Ornstein-Uhlenbeck semigroup following an approach due to Ledoux

The miR-17-92 cluster counteracts quiescence and chemoresistance in a distinct subpopulation of pancreatic cancer stem cells

Objective Cancer stem cells (CSCs) represent the root of many solid cancers including pancreatic ductal adenocarcinoma, are highly chemoresistant and represent the cellular source for disease relapse. However the mechanisms involved in these processes still need to be fully elucidated. Understanding the mechanisms implicated in chemoresistance and metastasis of pancreatic cancer is critical to improving patient outcomes. Design Micro-RNA (miRNA) expression analyses were performed to identify functionally defining epigenetic signatures in pancreatic CSC-enriched sphere-derived cells and gemcitabine-resistant pancreatic CSCs. Results We found the miR-17-92 cluster to be downregulated in chemoresistant CSCs versus non-CSCs and demonstrate its crucial relevance for CSC biology. In particular, overexpression of miR-17-92 reduced CSC self-renewal capacity, in vivo tumourigenicity and chemoresistance by targeting multiple NODAL/ACTIVIN/TGF-beta 1 signalling cascade members as well as directly inhibiting the downstream targets p21, p57 and TBX3. Overexpression of miR-17-92 translated into increased CSC proliferation and their eventual exhaustion via downregulation of p21 and p57. Finally, the translational impact of our findings could be confirmed in preclinical models for pancreatic cancer. Conclusions Our findings therefore identify the miR-17-92 cluster as a functionally determining family of miRNAs in CSCs, and highlight the putative potential of developing modulators of this cluster to overcome drug resistance in pancreatic CSCs.CH: ERC Advanced Investigator Grant (Pa-CSC 233460), European Community's Seventh Framework Programme (FP7/2007-2013) under grant agreement No 256974 (EPC-TM-NET) and No 602783 (CAM-PaC), the Subdireccion General de Evaluacion y Fomento de la Investigacion, Fondo de Investigacion Sanitaria (PS09/02129 \& PI12/02643), and the Programa Nacional de Internacionalizacion de la I+D, Subprogramma: FCCI 2009 (PLE2009-0105; Ministerio de Economia y Competitividad, Spain). MC: La Caixa Predoctoral Fellowship.S

XIPE: the X-ray Imaging Polarimetry Explorer

X-ray polarimetry, sometimes alone, and sometimes coupled to spectral and temporal variability measurements and to imaging, allows a wealth of physical phenomena in astrophysics to be studied. X-ray polarimetry investigates the acceleration process, for example, including those typical of magnetic reconnection in solar flares, but also emission in the strong magnetic fields of neutron stars and white dwarfs. It detects scattering in asymmetric structures such as accretion disks and columns, and in the so-called molecular torus and ionization cones. In addition, it allows fundamental physics in regimes of gravity and of magnetic field intensity not accessible to experiments on the Earth to be probed. Finally, models that describe fundamental interactions (e.g. quantum gravity and the extension of the Standard Model) can be tested. We describe in this paper the X-ray Imaging Polarimetry Explorer (XIPE), proposed in June 2012 to the first ESA call for a small mission with a launch in 2017 but not selected. XIPE is composed of two out of the three existing JET-X telescopes with two Gas Pixel Detectors (GPD) filled with a He-DME mixture at their focus and two additional GPDs filled with pressurized Ar-DME facing the sun. The Minimum Detectable Polarization is 14 % at 1 mCrab in 10E5 s (2-10 keV) and 0.6 % for an X10 class flare. The Half Energy Width, measured at PANTER X-ray test facility (MPE, Germany) with JET-X optics is 24 arcsec. XIPE takes advantage of a low-earth equatorial orbit with Malindi as down-link station and of a Mission Operation Center (MOC) at INPE (Brazil).Comment: 49 pages, 14 figures, 6 tables. Paper published in Experimental Astronomy http://link.springer.com/journal/1068