Asymptotic profiles for a nonlinear Kirchhoff equation with combined powers nonlinearity

We study asymptotic behavior of positive ground state solutions of the nonlinear Kirchhoff equation $$-\Big(a+b\int_{\mathbb R^N}|\nabla u|^2\Big)\Delta u+ \lambda u= u^{q-1}+ u^{p-1} \quad {\rm in} \ \mathbb R^N,$$ as $\lambda\to 0$ and $\lambda\to +\infty$, where $N=3$ or $N= 4$, $2<q\le p\le 2^*$, $2^*=\frac{2N}{N-2}$ is the Sobolev critical exponent, $a>0$, $b\ge 0$ are constants and $\lambda>0$ is a parameter. In particular, we prove that in the case $2<q<p=2^*$, as $\lambda\to 0$, after a suitable rescaling the ground state solutions of the problem converge to the unique positive solution of the equation $-\Delta u+u=u^{q-1}$ and as $\lambda\to +\infty$, after another rescaling the ground state solutions of the problem converge to a particular solution of the critical Emden-Fowler equation $-\Delta u=u^{2^*-1}$. We establish a sharp asymptotic characterisation of such rescalings, which depends in a non-trivial way on the space dimension $N=3$ and $N= 4$. We also discuss a connection of our results with a mass constrained problem associated to the Kirchhoff equation with the mass normalization constraint $\int_{\mathbb R^N}|u|^2=c^2$.Comment: 40 page

Nonexistence and optimal decay of supersolutions to Choquard equations in exterior domains

We consider a semilinear elliptic problem with a nonlinear term which is the product of a power and the Riesz potential of a power. This family of equations includes the Choquard or nonlinear Schroedinger--Newton equation. We show that for some values of the parameters the equation does not have nontrivial nonnegative supersolutions in exterior domains. The same techniques yield optimal decay rates when supersolutions exists.Comment: 47 pages, 8 figure

Don't work. Can't work? Why it's time to rethink security warnings

As the number of Internet users has grown, so have the security threats that they face online. Security warnings are one key strategy for trying to warn users about those threats; but recently, it has been questioned whether they are effective. We conducted a study in which 120 participants brought their own laptops to a usability test of a new academic article summary tool. They encountered a PDF download warning for one of the papers. All participants noticed the warning, but 98 (81.7%) downloaded the PDF file that triggered it. There was no significant difference between responses to a brief generic warning, and a longer specific one. The participants who heeded the warning were overwhelmingly female, and either had previous experience with viruses or lower levels of computing skills. Our analysis of the reasons for ignoring warnings shows that participants have become desensitised by frequent exposure and false alarms, and think they can recognise security risks. At the same time, their answers revealed some misunderstandings about security threats: for instance, they rely on anti-virus software to protect them from a wide range of threats, and do not believe that PDF files can infect their machine with viruses. We conclude that security warnings in their current forms are largely ineffective, and will remain so, unless the number of false positives can be reduced

Multiple solutions to a magnetic nonlinear Choquard equation

We consider the stationary nonlinear magnetic Choquard equation [(-\mathrm{i}\nabla+A(x))^{2}u+V(x)u=(\frac{1}{|x|^{\alpha}}\ast |u|^{p}) |u|^{p-2}u,\quad x\in\mathbb{R}^{N}%] where $A\$is a real valued vector potential, $V$ is a real valued scalar potential$,$ $N\geq3$, $\alpha\in(0,N)$ and $2-(\alpha/N) <p<(2N-\alpha)/(N-2)$. \ We assume that both $A$ and $V$ are compatible with the action of some group $G$ of linear isometries of $\mathbb{R}^{N}$. We establish the existence of multiple complex valued solutions to this equation which satisfy the symmetry condition $$u(gx)=\tau(g)u(x)\text{\ \ \ for all}g\in G,\text{}x\in\mathbb{R}^{N},$$ where $\tau:G\rightarrow\mathbb{S}^{1}$ is a given group homomorphism into the unit complex numbers.Comment: To appear on ZAM

Critical solutions in topologically gauged N=8 CFTs in three dimensions

In this paper we discuss some special (critical) background solutions that arise in topological gauged ${\mathcal N}=8$ three-dimensional CFTs with SO(N) gauge group. These solutions solve the TMG equations (containing the parameters $\mu$ and $l$) for a certain set of values of $\mu l$ obtained by varying the number of scalar fields with a VEV. Apart from Minkowski, chiral round $AdS_3$ and null-warped $AdS_3$ (or Schr\"odinger(z=2)) we identify also a more exotic solution recently found in $TMG$ by Ertl, Grumiller and Johansson. We also discuss the spectrum, symmetry breaking pattern and the supermultiplet structure in the various backgrounds and argue that some properties are due to their common origin in a conformal phase. Some of the scalar fields, including all higgsed ones, turn out to satisfy three-dimensional singleton field equations. Finally, we note that topologically gauged ${\mathcal N}=6$ ABJ(M) theories have a similar, but more restricted, set of background solutions.Comment: 34 pages, v2: minor corrections, note about a new solution added in final section, v3: two footnotes adde

Casimir energy of massive MIT fermions in a Bohm-Aharonov background

We study the effect of a background flux string on the vacuum energy of massive Dirac fermions in 2+1 dimensions confined to a finite spatial region through MIT boundary conditions. We treat two admissible self-adjoint extensions of the Hamiltonian and compare the results. In particular, for one of these extensions, the Casimir energy turns out to be discontinuous at integer values of the flux.Comment: 16 pages, 3 figure

Drosophila Neurotrophins Reveal a Common Mechanism for Nervous System Formation

Neurotrophic interactions occur in Drosophila, but to date, no neurotrophic factor had been found. Neurotrophins are the main vertebrate secreted signalling molecules that link nervous system structure and function: they regulate neuronal survival, targeting, synaptic plasticity, memory and cognition. We have identified a neurotrophic factor in flies, Drosophila Neurotrophin (DNT1), structurally related to all known neurotrophins and highly conserved in insects.By investigating with genetics the consequences of removing DNT1 or adding it in excess, we show that DNT1 maintains neuronal survival, as more neurons die in DNT1 mutants and expression of DNT1 rescues naturally occurring cell death, and it enables targeting by motor neurons. We show that Spa¨ tzle and a further fly neurotrophin superfamily member, DNT2, also have neurotrophic functions in flies. Our findings imply that most likely a neurotrophin was present in the common ancestor of all bilateral organisms, giving rise to invertebrate and vertebrate neurotrophins through gene or whole-genome duplications. This work provides a missing link between aspects of neuronal function in flies and vertebrates, and it opens the opportunity to use Drosophila to investigate further aspects of neurotrophin function and to model related diseases

Clinical and morphological characteristics of head-facial haemangiomas

BACKGROUND: Haemangiomas of the head or face are a frequent vascular pathology, consisting in an embryonic dysplasia that involves the cranial-facial vascular network. Haemangiomas show clinical, morphological, developmental and structural changes during their course. METHODS: The clinical characteristics of head-facial haemagiomas were studied in 28 individuals (9 males and 19 females) admitted in our Hospital. Sixteen of these patients(n = 16) underwent surgery for the removal of the haemangiomas. All the removed tissues were transferred in experimental laboratories for the staining of microanatomical details, somatic and visceral nerve fibres, adrenergic and catecholaminergic nerve fibres. Beta-adrenergic receptors were died with a fluorescent method. All results were submitted to the quantitative analysis of images and statistical evaluation of the data. RESULTS: The morphological results revealed numerous micro-anatomical characteristics of the haemangiomatous vessels. The somatic and visceral nerve fibres were poor and located exclusively in the adventitial layer. There was a marked decrease of adrenergic nerve fibres in the haemangiomatous vessels. The fluorescence of catecholaminergic nerve fibres and the overall area of fluorescent structures were also decreased in haemangiomatous vessels. Beta adrenergic receptors are strongly decreased in haemangiomatous vessels. The morphometrical analysis of images and statistical evaluation of the data confirmed all our experimental results. CONCLUSION: The catecholaminergic innervation of the human haemangiomatous vessels comprises nerve fibres containing the main catecholaminergic neurotransmitters that are sympathetic in nature. These neurotransmitters are closely related to beta-adrenergic receptors. The sympathetic nervous system plays a key role in the control of the vascular bed flow and vascular motility in both normal and haemangiomatous vessels

Overview of the techniques used for the study of non-terrestrial bodies: Proposition of novel non-destructive methodology

Meteorites and impact glasses have been largely analysed using different techniques, but most studies have been focused on their geologicalemineralogical characterization and isotopic ratios, mainly of a destructive nature. However, much more information can be gained by applying novel non-destructive analytical procedures and techniques that have been scarcely used to analyse these materials. This overview presents some new methodologies to study these materials and compares these new approaches with the commonly used ones. Techniques such as X-Ray Fluorescence (XRF) and Laser Induced Breakdown Spectroscopy (LIBS), for elemental characterization, the hyphenated Raman spectroscopy- SEM/EDS and the combination of them, allow extracting simultaneous information from elemental, molecular and structural data of the studied sample; furthermore, the spectroscopic image capabilities of such techniques allow a better understanding of the mineralogical distribution. © 2017 Elsevier B.V. All rights reserved.Ministerio de Economía, Industria y Competitividad (project ESP2014-56138-C3-2-R