A three solutions theorem for Pucci's extremal operator and its application

In this article we prove a three solution type theorem for the following boundary value problem: \begin{equation*} \label{abs} \begin{cases} -\mathcal{M}_{\lambda,\Lambda}^+(D^2u) =f(u)& \text{in }\Omega,\\ u =0& \text{on }\partial\Omega, \end{cases} \end{equation*} where $\Omega$ is a bounded smooth domain in $\mathbb{R}^N$ and $f\colon [0,\infty]\to[0,\infty]$ is a $C^{\alpha}$ function. This is motivated by the work of Amann \cite{aman} and Shivaji \cite{shivaji1987remark}, where a three solutions theorem has been established for the Laplace operator. Furthermore, using this result we show the existence of three positive solutions to above boundary value by explicitly constructing two ordered pairs of sub and supersolutions when $f$ has a sublinear growth and $f(0)=0.$

The supernova remnant populations of the galaxies NGC 45, NGC 55, NGC 1313, NGC 7793: luminosity and excitation functions

ABSTRACT We present a systematic study of the supernova remnant (SNR) populations in the nearby galaxies NGC 45, NGC 55, NGC 1313, and NGC 7793 based on deep H $\rm {\alpha }$ and [S ii] imaging. We find 42 candidate and 51 possible candidate SNRs based on the [S ii]/H $\rm {\alpha }$>0.4 criterion, 81 of which are new identifications. We derive the H $\rm {\alpha }$ and the joint [S ii]–H $\rm {\alpha }$ luminosity functions after accounting for incompleteness effects. We find that the H $\rm {\alpha }$ luminosity function of the overall sample is described with a skewed Gaussian with a mean equal to $\rm \log (L_{H\alpha }/10^{36}\, erg\, s^{-1})=0.07$ and $\rm \sigma (\log (L_{H\alpha }/10^{36}\, erg\, s^{-1}))=0.58$. The joint [S ii]–H $\rm {\alpha }$ function is parametrized by a skewed Gaussian along the log([S ii]$\rm /10^{36}\, erg\, s^{-1}) = 0.88 \times \log (L_{H\alpha }/10^{36}\, erg\, s^{-1}) - 0.06$ line and a truncated Gaussian with $\rm \mu (\log (L_{[S\, II]}/10^{36})) = 0.024$ and $\rm \sigma (\log (L_{[S\, II]}/10^{36})) = 0.14$, on its vertical direction. We also define the excitation function as the number density of SNRs as a function of their [S ii]/H $\rm {\alpha }$ ratios. This function is represented by a truncated Gaussian with a mean at −0.014. We find a sub-linear [S ii]–H $\rm {\alpha }$ relation indicating lower excitation for the more luminous objects.

Weak KAM Theory and Forcing Equivalence

This chapter describes weak Kolmogorov-Arnold-Moser (KAM) theory and forcing relation. One change from the standard presentation is that one needs to modify the definition of Tonelli Hamiltonians to allow different periods in the t component. The chapter points out an alternative definition of the alpha function, namely, one can replace the class of minimal measures with the class of closed measures. It then considers a dual setting which corresponds to forward dynamic. It also looks at elementary solutions, static classes, and Peierls barrier. In many parts of the proof, the chapter studies the hyperbolic property of a minimizing orbit, for which the concept of Green bundles is very useful.

From Bion to Delany

Literature furnishes a particular vertex to see reality through narrative fiction. In particular, science fiction literature, which creates a fantastic situation starting from realistic data (history, science, cultures), may be considered a kind of creative process. It uses heterogeneous things and “integrates” them into a homogeneous, new and comprehensible product. Science fiction writing allows the objects of the real to be reprocessed in terms which are thinkable at the current moment. Using the terminology established in psychoanalysis by Wilfred Bion this reprocessing work is a transformation. According to Bion we can hypothesise that the writer of the science fiction literary work serves as a “container” and the science fiction novel, considered a different way to represent reality and not just a simple editorial product, serves as a alpha-function to make concepts that were not previously thinkable or understandable. Between the 70s and the 80s the writer Samuel Delany theorized and put into practice the use of a literary model called “modular calculus”. This model allows the literary work of making something unthinkable into thinkable. The purpose of this paper is to highlight how modular calculus is a particular type of Bionian transformation, and how the science fiction novel can play the role of alpha-function, transforming unthinkable concepts into thinkable ones.

Targeting HIF-1α Function in Cancer through the Chaperone Action of NQO1

HIF-1α is a master regulator of oxygen homeostasis involved in different stages of cancer development. Thus, HIF-1α inhibition represents an interesting target for anti-cancer therapy. It was recently shown that HIF-1α interaction with NQO1 inhibits its proteasomal degradation, thus suggesting that targeting the stability of NQO1 could led to destabilization of HIF-1α as a therapeutic approach. Since the molecular interactions of NQO1 with HIF-1α are beginning to be unraveled, we review here our current knowledge on the intracellular functions and stability of NQO1, its pharmacological modulation by small ligands, and the molecular determinants of its roles as a chaperone of many different proteins including cancer-associated factors such as p53 and p73α. This knowledge is then discussed in the context of potentially targeting the intracellular stability of HIF-1α by acting on its chaperone, NQO1. This could result in novel anti-cancer therapies.