124,018 research outputs found
On the number of rich lines in truly high dimensional sets
We prove a new upper bound on the number of -rich lines (lines with at
least points) in a `truly' -dimensional configuration of points
. More formally, we show that, if the number
of -rich lines is significantly larger than then there must exist
a large subset of the points contained in a hyperplane. We conjecture that the
factor can be replaced with a tight . If true, this would
generalize the classic Szemer\'edi-Trotter theorem which gives a bound of
on the number of -rich lines in a planar configuration. This
conjecture was shown to hold in in the seminal work of Guth and
Katz \cite{GK10} and was also recently proved over (under some
additional restrictions) \cite{SS14}. For the special case of arithmetic
progressions ( collinear points that are evenly distanced) we give a bound
that is tight up to low order terms, showing that a -dimensional grid
achieves the largest number of -term progressions.
The main ingredient in the proof is a new method to find a low degree
polynomial that vanishes on many of the rich lines. Unlike previous
applications of the polynomial method, we do not find this polynomial by
interpolation. The starting observation is that the degree Veronese
embedding takes -collinear points to linearly dependent images. Hence,
each collinear -tuple of points, gives us a dependent -tuple of images.
We then use the design-matrix method of \cite{BDWY12} to convert these 'local'
linear dependencies into a global one, showing that all the images lie in a
hyperplane. This then translates into a low degree polynomial vanishing on the
original set
Point-curve incidences in the complex plane
We prove an incidence theorem for points and curves in the complex plane.
Given a set of points in and a set of curves with
degrees of freedom, Pach and Sharir proved that the number of point-curve
incidences is . We
establish the slightly weaker bound
on the number of incidences between points and (complex) algebraic
curves in with degrees of freedom. We combine tools from
algebraic geometry and differential geometry to prove a key technical lemma
that controls the number of complex curves that can be contained inside a real
hypersurface. This lemma may be of independent interest to other researchers
proving incidence theorems over .Comment: The proof was significantly simplified, and now relies on the
Picard-Lindelof theorem, rather than on foliation
Exploration of Parameter Spaces in a Virtual Observatory
Like every other field of intellectual endeavor, astronomy is being
revolutionised by the advances in information technology. There is an ongoing
exponential growth in the volume, quality, and complexity of astronomical data
sets, mainly through large digital sky surveys and archives. The Virtual
Observatory (VO) concept represents a scientific and technological framework
needed to cope with this data flood. Systematic exploration of the observable
parameter spaces, covered by large digital sky surveys spanning a range of
wavelengths, will be one of the primary modes of research with a VO. This is
where the truly new discoveries will be made, and new insights be gained about
the already known astronomical objects and phenomena. We review some of the
methodological challenges posed by the analysis of large and complex data sets
expected in the VO-based research. The challenges are driven both by the size
and the complexity of the data sets (billions of data vectors in parameter
spaces of tens or hundreds of dimensions), by the heterogeneity of the data and
measurement errors, including differences in basic survey parameters for the
federated data sets (e.g., in the positional accuracy and resolution,
wavelength coverage, time baseline, etc.), various selection effects, as well
as the intrinsic clustering properties (functional form, topology) of the data
distributions in the parameter spaces of observed attributes. Answering these
challenges will require substantial collaborative efforts and partnerships
between astronomers, computer scientists, and statisticians.Comment: Invited review, 10 pages, Latex file with 4 eps figures, style files
included. To appear in Proc. SPIE, v. 4477 (2001
Quantitative chemical tagging, stellar ages and the chemo-dynamical evolution of the Galactic disc
The early science results from the new generation of high-resolution stellar
spectroscopic surveys, such as GALAH and the Gaia-ESO survey, will represent
major milestones in the quest to chemically tag the Galaxy. Yet this technique
to reconstruct dispersed coeval stellar groups has remained largely untested
until recently. We build on previous work that developed an empirical chemical
tagging probability function, which describes the likelihood that two field
stars are conatal, that is, they were formed in the same cluster environment.
In this work we perform the first ever blind chemical tagging experiment, i.e.,
tagging stars with no known or otherwise discernable associations, on a sample
of 714 disc field stars with a number of high quality high resolution
homogeneous metal abundance measurements. We present evidence that chemical
tagging of field stars does identify coeval groups of stars, yet these groups
may not represent distinct formation sites, e.g. as in dissolved open clusters,
as previously thought. Our results point to several important conclusions,
among them that group finding will be limited strictly to chemical abundance
space, e.g. stellar ages, kinematics, colors, temperature and surface gravity
do not enhance the detectability of groups. We also demonstrate that in
addition to its role in probing the chemical enrichment and kinematic history
of the Galactic disc, chemical tagging represents a powerful new stellar age
determination technique.Comment: 12 pages, 9 figures, accepted for publication in Monthly Notices of
the Royal Astronomical Society (MNRAS
: A Unique Mott Hubbard Insulator
We discuss the recently discovered system , a realization
of an exactly solvable model proposed two decades earlier. We propose its
interpretation as a Mott Hubbard insulator. The possible superconducting phase
arising from doping is explored, and its nature as well as its importance for
testing the RVB theory of superconductivity are discussed.Comment: 18 pages,7 figures, Based on Invited Talk at 16th Nishinomiya Yukawa
memorial symposium, Nov 13-14, Nishinomiya. To appear in Progress in
Theoretical Physics, Suppliment 145(2002
Type-Ia Supernova Remnant Shell At Seen In The Three Sightlines Toward The Gravitationally Lensed Qso B1422+231
Using the Subaru 8.2m Telescope with an IRCS Echelle spectrograph, we
obtained high-resolution (R=10,000) near-infrared (1.01-1.38 \mu m) spectra of
images A and B of the gravitationally lensed QSO B1422+231 (z=3.628) consisting
of four known lensed images. We detected MgII absorption lines at z=3.54, which
show a large variance of column densities (~ 0.3 dex) and velocities (~ 10
km/s) between the sightlines A and B with a projected separation of only
8.4h_{70}^{-1} pc at the redshift. This is the smallest spatial structure of
the high-z gas clouds ever detected after Rauch et al. found a 20-pc scale
structure for the same z=3.54 absorption system using optical spectra of images
A and C. The observed systematic variances imply that the system is an
expanding shell as originally suggested by Rauch et al. By combining the data
for three sightlines, we managed to constrain the radius and expansion velocity
of the shell (~ 50-100 pc, 130 km/s), concluding that the shell is truly a
supernova remnant (SNR) rather than other types of shell objects, such as a
giant HII region. We also detected strong FeII absorption lines for this
system, but with much broader Doppler width than that of \alpha-element lines.
We suggest that this FeII absorption line originates in a localized FeII-rich
gas cloud that is not completely mixed with plowed ambient interstellar gas
clouds showing other \alpha-element low-ion absorption lines. Along with the Fe
richness, we conclude that the SNR is produced by an SNIa explosion.Comment: 16 pages, 15 figures, to be published in The Astrophysical Journa
Hot bottom burning and s-process nucleosynthesis in massive AGB stars at the beginning of the thermally-pulsing phase
We report the first spectroscopic identification of massive Galactic
asymptotic giant branch (AGB) stars at the beginning of the thermal pulse (TP)
phase. These stars are the most Li-rich massive AGBs found to date, super
Li-rich AGBs with logE(Li)~3-4. The high Li overabundances are accompanied by
weak or no s-process element (i.e. Rb and Zr) enhancements. A comparison of our
observations with the most recent hot bottom burning (HBB) and s-process
nucleosynthesis models confirms that HBB is strongly activated during the first
TPs but the 22Ne neutron source needs many more TP and third dredge-up episodes
to produce enough Rb at the stellar surface. We also show that the short-lived
element Tc, usually used as an indicator of AGB genuineness, is not detected in
massive AGBs which is in agreement with the theoretical predictions when the
22Ne neutron source dominates the s-process nucleosynthesis.Comment: Accepted for publication in Astronomy & Astrophysics Letters (7
pages, 5 figures and 1 table); final version (language corrected
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