569 research outputs found
Connectivity and tree structure in finite graphs
Considering systems of separations in a graph that separate every pair of a
given set of vertex sets that are themselves not separated by these
separations, we determine conditions under which such a separation system
contains a nested subsystem that still separates those sets and is invariant
under the automorphisms of the graph.
As an application, we show that the -blocks -- the maximal vertex sets
that cannot be separated by at most vertices -- of a graph live in
distinct parts of a suitable tree-decomposition of of adhesion at most ,
whose decomposition tree is invariant under the automorphisms of . This
extends recent work of Dunwoody and Kr\"on and, like theirs, generalizes a
similar theorem of Tutte for .
Under mild additional assumptions, which are necessary, our decompositions
can be combined into one overall tree-decomposition that distinguishes, for all
simultaneously, all the -blocks of a finite graph.Comment: 31 page
Eigenvalue Bounds for Perturbations of Schrodinger Operators and Jacobi Matrices With Regular Ground States
We prove general comparison theorems for eigenvalues of perturbed Schrodinger
operators that allow proof of Lieb--Thirring bounds for suitable non-free
Schrodinger operators and Jacobi matrices.Comment: 11 page
Lieb-Thirring inequalities for geometrically induced bound states
We prove new inequalities of the Lieb-Thirring type on the eigenvalues of
Schr\"odinger operators in wave guides with local perturbations. The estimates
are optimal in the weak-coupling case. To illustrate their applications, we
consider, in particular, a straight strip and a straight circular tube with
either mixed boundary conditions or boundary deformations.Comment: LaTeX2e, 14 page
Toward a Broadband Astro-comb: Effects of Nonlinear Spectral Broadening in Optical Fibers
We propose and analyze a new approach to generate a broadband astro-comb by
spectral broadening of a narrowband astro-comb inside a highly nonlinear
optical fiber. Numerical modeling shows that cascaded four-wave-mixing
dramatically degrades the input comb's side-mode suppression and causes
side-mode amplitude asymmetry. These two detrimental effects can systematically
shift the center-of-gravity of astro-comb spectral lines as measured by an
astrophysical spectrograph with resolution \approx100,000; and thus lead to
wavelength calibration inaccuracy and instability. Our simulations indicate
that this performance penalty, as a result of nonlinear spectral broadening,
can be compensated by using a filtering cavity configured for double-pass. As
an explicit example, we present a design based on an Yb-fiber source comb (with
1 GHz repetition rate) that is filtered by double-passing through a low finesse
cavity (finesse = 208), and subsequent spectrally broadened in a 2-cm,
SF6-glass photonic crystal fiber. Spanning more than 300 nm with 16 GHz line
spacing, the resulting astro-comb is predicted to provide 1 cm/s (~10 kHz)
radial velocity calibration accuracy for an astrophysical spectrograph. Such
extreme performance will be necessary for the search for and characterization
of Earth-like extra-solar planets, and in direct measurements of the change of
the rate of cosmological expansion.Comment: 9 pages, 6 figure
Upper and lower limits on the number of bound states in a central potential
In a recent paper new upper and lower limits were given, in the context of
the Schr\"{o}dinger or Klein-Gordon equations, for the number of S-wave
bound states possessed by a monotonically nondecreasing central potential
vanishing at infinity. In this paper these results are extended to the number
of bound states for the -th partial wave, and results are also
obtained for potentials that are not monotonic and even somewhere positive. New
results are also obtained for the case treated previously, including the
remarkably neat \textit{lower} limit with (valid in the Schr\"{o}dinger case, for a class of potentials
that includes the monotonically nondecreasing ones), entailing the following
\textit{lower} limit for the total number of bound states possessed by a
monotonically nondecreasing central potential vanishing at infinity: N\geq
\{\{(\sigma+1)/2\}\} {(\sigma+3)/2\} \}/2 (here the double braces denote of
course the integer part).Comment: 44 pages, 5 figure
(De)Localization in the Prime Schrodinger Operator
It is reported a combined numerical approach to study the localization
properties of the one-dimensional tight-binding model with potential modulated
along the prime numbers. A localization-delocalization transition was found as
function of the potential intensity; it is also argued that there are
delocalized states for any value of the potential intensity.Comment: 7 pages, 4 figures; to be published in J. Phys. A: Math. Ge
Inferring statistics of planet populations by means of automated microlensing searches
(abridged) The study of other worlds is key to understanding our own, and not
only provides clues to the origin of our civilization, but also looks into its
future. Rather than in identifying nearby systems and learning about their
individual properties, the main value of the technique of gravitational
microlensing is in obtaining the statistics of planetary populations within the
Milky Way and beyond. Only the complementarity of different techniques
currently employed promises to yield a complete picture of planet formation
that has sufficient predictive power to let us understand how habitable worlds
like ours evolve, and how abundant such systems are in the Universe. A
cooperative three-step strategy of survey, follow-up, and anomaly monitoring of
microlensing targets, realized by means of an automated expert system and a
network of ground-based telescopes is ready right now to be used to obtain a
first census of cool planets with masses reaching even below that of Earth
orbiting K and M dwarfs in two distinct stellar populations, namely the
Galactic bulge and disk. The hunt for extra-solar planets acts as a principal
science driver for time-domain astronomy with robotic-telescope networks
adopting fully-automated strategies. Several initiatives, both into facilities
as well as into advanced software and strategies, are supposed to see the
capabilities of gravitational microlensing programmes step-wise increasing over
the next 10 years. New opportunities will show up with high-precision
astrometry becoming available and studying the abundance of planets around
stars in neighbouring galaxies becoming possible. Finally, we should not miss
out on sharing the vision with the general public, and make its realization to
profit not only the scientists but all the wider society.Comment: 10 pages in PDF format. White paper submitted to ESA's Exo-Planet
Roadmap Advisory Team (EPR-AT); typos corrected. The embedded figures are
available from the author on request. See also "Towards A Census of
Earth-mass Exo-planets with Gravitational Microlensing" by J.P. Beaulieu, E.
Kerins, S. Mao et al. (arXiv:0808.0005
Sufficient conditions for two-dimensional localization by arbitrarily weak defects in periodic potentials with band gaps
We prove, via an elementary variational method, 1d and 2d localization within
the band gaps of a periodic Schrodinger operator for any mostly negative or
mostly positive defect potential, V, whose depth is not too great compared to
the size of the gap. In a similar way, we also prove sufficient conditions for
1d and 2d localization below the ground state of such an operator. Furthermore,
we extend our results to 1d and 2d localization in d dimensions; for example, a
linear or planar defect in a 3d crystal. For the case of D-fold degenerate band
edges, we also give sufficient conditions for localization of up to D states.Comment: 9 pages, 3 figure
A Centre-Stable Manifold for the Focussing Cubic NLS in
Consider the focussing cubic nonlinear Schr\"odinger equation in : It admits special solutions of the form
, where is a Schwartz function and a positive
() solution of The space of
all such solutions, together with those obtained from them by rescaling and
applying phase and Galilean coordinate changes, called standing waves, is the
eight-dimensional manifold that consists of functions of the form . We prove that any solution starting
sufficiently close to a standing wave in the norm and situated on a certain codimension-one local
Lipschitz manifold exists globally in time and converges to a point on the
manifold of standing waves. Furthermore, we show that \mc N is invariant
under the Hamiltonian flow, locally in time, and is a centre-stable manifold in
the sense of Bates, Jones. The proof is based on the modulation method
introduced by Soffer and Weinstein for the -subcritical case and adapted
by Schlag to the -supercritical case. An important part of the proof is
the Keel-Tao endpoint Strichartz estimate in for the nonselfadjoint
Schr\"odinger operator obtained by linearizing around a standing wave solution.Comment: 56 page
I. Flux and color variations of the quadruply imaged quasar HE 0435-1223
aims: We present VRi photometric observations of the quadruply imaged quasar
HE 0435-1223, carried out with the Danish 1.54m telescope at the La Silla
Observatory. Our aim was to monitor and study the magnitudes and colors of each
lensed component as a function of time. methods: We monitored the object during
two seasons (2008 and 2009) in the VRi spectral bands, and reduced the data
with two independent techniques: difference imaging and PSF (Point Spread
Function) fitting.results: Between these two seasons, our results show an
evident decrease in flux by ~0.2-0.4 magnitudes of the four lensed components
in the three filters. We also found a significant increase (~0.05-0.015) in
their V-R and R-i color indices. conclusions: These flux and color variations
are very likely caused by intrinsic variations of the quasar between the
observed epochs. Microlensing effects probably also affect the brightest "A"
lensed component.Comment: 10 pages, 8 figure
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