On the H\'enon-Lane-Emden conjecture

We consider Liouville-type theorems for the following H\'{e}non-Lane-Emden system \hfill -\Delta u&=& |x|^{a}v^p \text{in} \mathbb{R}^N, \hfill -\Delta v&=& |x|^{b}u^q \text{in} \mathbb{R}^N, when $pq>1$, $p,q,a,b\ge0$. The main conjecture states that there is no non-trivial non-negative solution whenever $(p,q)$ is under the critical Sobolev hyperbola, i.e. $\frac{N+a}{p+1}+\frac{N+b}{q+1}>{N-2}$. We show that this is indeed the case in dimension N=3 provided the solution is also assumed to be bounded, extending a result established recently by Phan-Souplet in the scalar case. Assuming stability of the solutions, we could then prove Liouville-type theorems in higher dimensions. For the scalar cases, albeit of second order ($a=b$ and $p=q$) or of fourth order ($a\ge 0=b$ and $p>1=q$), we show that for all dimensions $N\ge 3$ in the first case (resp., $N\ge 5$ in the second case), there is no positive solution with a finite Morse index, whenever $p$ is below the corresponding critical exponent, i.e $1<p<\frac{N+2+2a}{N-2}$ (resp., $1<p<\frac{N+4+2a}{N-4}$). Finally, we show that non-negative stable solutions of the full H\'{e}non-Lane-Emden system are trivial provided \label{sysdim00} N<2+2(\frac{p(b+2)+a+2}{pq-1}) (\sqrt{\frac{pq(q+1)}{p+1}}+ \sqrt{\frac{pq(q+1)}{p+1}-\sqrt\frac{pq(q+1)}{p+1}}).Comment: Theorem 4 has been added in the new version. 23 pages, Comments are welcome. Updated version - if any - can be downloaded at http://www.birs.ca/~nassif/ or http://www.math.ubc.ca/~fazly/research.htm

Global well-posedness and scattering for the defocusing energy-critical nonlinear Schr\"odinger equation in R1+4\R^{1+4}

We obtain global well-posedness, scattering, uniform regularity, and global $L^6_{t,x}$ spacetime bounds for energy-space solutions to the defocusing energy-critical nonlinear Schr\"odinger equation in $\R\times\R^4$. Our arguments closely follow those of Colliander-Keel-Staffilani-Takaoka-Tao, though our derivation of the frequency-localized interaction Morawetz estimate is somewhat simpler. As a consequence, our method yields a better bound on the $L^6_{t,x}$-norm

The support of the logarithmic equilibrium measure on sets of revolution in R3\R^3

For surfaces of revolution $B$ in $\R^3$, we investigate the limit distribution of minimum energy point masses on $B$ that interact according to the logarithmic potential $\log (1/r)$, where $r$ is the Euclidean distance between points. We show that such limit distributions are supported only on the ``out-most'' portion of the surface (e.g., for a torus, only on that portion of the surface with positive curvature). Our analysis proceeds by reducing the problem to the complex plane where a non-singular potential kernel arises whose level lines are ellipses

Non-Abelian Proca model based on the improved BFT formalism

We present the newly improved Batalin-Fradkin-Tyutin (BFT) Hamiltonian formalism and the generalization to the Lagrangian formulation, which provide the much more simple and transparent insight to the usual BFT method, with application to the non-Abelian Proca model which has been an difficult problem in the usual BFT method. The infinite terms of the effectively first class constraints can be made to be the regular power series forms by ingenious choice of $X_{\alpha \beta}$ and $\omega^{\alpha \beta}$-matrices. In this new method, the first class Hamiltonian, which also needs infinite correction terms is obtained simply by replacing the original variables in the original Hamiltonian with the BFT physical variables. Remarkably all the infinite correction terms can be expressed in the compact exponential form. We also show that in our model the Poisson brackets of the BFT physical variables in the extended phase space are the same structure as the Dirac brackets of the original phase space variables. With the help of both our newly developed Lagrangian formulation and Hamilton's equations of motion, we obtain the desired classical Lagrangian corresponding to the first class Hamiltonian which can be reduced to the generalized St\"uckelberg Lagrangian which is non-trivial conjecture in our infinitely many terms involved in Hamiltonian and Lagrangian.Comment: Notable improvements in Sec. I

Complete Constant Mean Curvature surfaces and Bernstein type Theorems in M2×R\mathbb{M}^2\times \mathbb{R}

In this paper we study constant mean curvature surfaces $\Sigma$ in a product space, $\mathbb{M}^2\times \mathbb{R}$, where $\mathbb{M}^2$ is a complete Riemannian manifold. We assume the angle function \nu = \meta{N}{\partial_t} does not change sign on $\Sigma$. We classify these surfaces according to the infimum $c(\Sigma)$ of the Gaussian curvature of the projection of $\Sigma$. When $H \neq 0$ and $c(\Sigma)\geq 0$, then $\Sigma$ is a cylinder over a complete curve with curvature 2H. If H=0 and $c(\Sigma) \geq 0$, then $\Sigma$ must be a vertical plane or $\Sigma$ is a slice $\mathbb{M}^2 \times {t}$, or $\mathbb{M}^2 \equiv \mathbb{R}^2$ with the flat metric and $\Sigma$ is a tilted plane (after possibly passing to a covering space). When $c(\Sigma)\sqrt{-c(\Sigma)} /2$, then $\Sigma$ is a vertical cylinder over a complete curve of $\mathbb{M}^2$ of constant geodesic curvature $2H$. This result is optimal. We also prove a non-existence result concerning complete multi-graphs in $\mathbb{M}^2\times \mathbb{R}$, when $c(\mathbb{M}^2)<0$

A population study of type II bursts in the Rapid Burster

Type II bursts are thought to arise from instabilities in the accretion flow onto a neutron star in an X-ray binary. Despite having been known for almost 40 years, no model can yet satisfactorily account for all their properties. To shed light on the nature of this phenomenon and provide a reference for future theoretical work, we study the entire sample of Rossi X-ray Timing Explorer data of type II bursts from the Rapid Burster (MXB 1730-335). We find that type II bursts are Eddington-limited in flux, that a larger amount of energy goes in the bursts than in the persistent emission, that type II bursts can be as short as 0.130 s, and that the distribution of recurrence times drops abruptly below 15-18 s. We highlight the complicated feedback between type II bursts and the NS surface thermonuclear explosions known as type I bursts, and between type II bursts and the persistent emission. We review a number of models for type II bursts. While no model can reproduce all the observed burst properties and explain the source uniqueness, models involving a gating role for the magnetic field come closest to matching the properties of our sample. The uniqueness of the source may be explained by a special combination of magnetic field strength, stellar spin period and alignment between the magnetic field and the spin axis.Comment: Accepted 2015 February 12. Received 2015 February 10; in original form 2014 December 1

Some selected simulation experiments with the European Commission's QUEST model

This paper presents a set of simulation experiments using the European Commission's QUEST model to evaluate the effects of policy impulses and permanent supply side shocks in the four major EU economies. The simulation analysis illustrates the transmission mechanisms of specific monetary and fiscal policy shocks as well as two examples of permanent supply shocks.QUEST model, supply side shocks, monetary and fiscal policy, R�ger, in 't Veld,

Achromatic late-time variability in thermonuclear X-ray bursts - an accretion disk disrupted by a nova-like shell?

An unusual Eddington-limited thermonuclear X-ray burst was detected from the accreting neutron star in 2S 0918-549 with the Rossi X-ray Timing Explorer. The burst commenced with a brief (40 ms) precursor and maintained near-Eddington fluxes during the initial 77 s. These characteristics are indicative of a nova-like expulsion of a shell from the neutron star surface. Starting 122 s into the burst, the burst shows strong (87 +/- 1% peak-to-peak amplitude) achromatic fluctuations for 60 s. We speculate that the fluctuations are due to Thompson scattering by fully-ionized inhomogeneities in a resettling accretion disk that was disrupted by the effects of super-Eddington fluxes. An expanding shell may be the necessary prerequisite for the fluctuations.Comment: 7 pages, 4 figures. Submitted to A&