37,060 research outputs found
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 ,
. The main conjecture states that there is no non-trivial
non-negative solution whenever is under the critical Sobolev hyperbola,
i.e. .
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 ( and ) or of fourth
order ( and ), we show that for all dimensions in the
first case (resp., in the second case), there is no positive solution
with a finite Morse index, whenever is below the corresponding critical
exponent, i.e (resp., ).
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
Complete Constant Mean Curvature surfaces and Bernstein type Theorems in
In this paper we study constant mean curvature surfaces in a product
space, , where is a complete
Riemannian manifold. We assume the angle function \nu = \meta{N}{\partial_t}
does not change sign on . We classify these surfaces according to the
infimum of the Gaussian curvature of the projection of .
When and , then is a cylinder over a
complete curve with curvature 2H. If H=0 and , then
must be a vertical plane or is a slice , or
with the flat metric and is a
tilted plane (after possibly passing to a covering space).
When , then is a vertical
cylinder over a complete curve of of constant geodesic curvature
. This result is optimal.
We also prove a non-existence result concerning complete multi-graphs in
, when
Global well-posedness and scattering for the defocusing energy-critical nonlinear Schr\"odinger equation in
We obtain global well-posedness, scattering, uniform regularity, and global
spacetime bounds for energy-space solutions to the defocusing
energy-critical nonlinear Schr\"odinger equation in . 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
-norm
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, Rïżœger, in 't Veld,
The support of the logarithmic equilibrium measure on sets of revolution in
For surfaces of revolution in , we investigate the limit
distribution of minimum energy point masses on that interact according to
the logarithmic potential , where 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 and -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
New BeppoSAX-WFC results on superbursts
Presently seven superbursters have been identified representing 10% of the
total Galactic X-ray burster population. Four superbursters were discovered
with the Wide Field Cameras (WFCs) on BeppoSAX and three with the All-Sky
Monitor and Proportional Counter Array on RXTE. We discuss the properties of
superbursters as derived from WFC observations. There are two interesting
conclusions. First, the average recurrence time of superbursts among X-ray
bursters that are more luminous than 10% of the Eddington limit is 1.5 yr per
object. Second, superbursters systematically have higher alpha values and
shorter ordinary bursts than most bursters that have not exhibited superbursts,
indicating a higher level of stable thermonuclear helium burning. Theory
predicts hitherto undetected superbursts from the most luminous neutron stars.
We investigate the prospects for finding these in GX~17+2.Comment: Submitted in January 2004 for the Proceedings of the meeting 'X-Ray
Timing 2003: Rossi and Beyond', eds. P. Kaaret, F. K. Lamb, & J. H. Swank
(Melville, NY: American Institute of Physics
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
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