1,135,303 research outputs found
Volume functions of linear series
The volume of a Cartier divisor is an asymptotic invariant, which measures
the rate of growth of sections of powers of the divisor. It extends to a
continuous, homogeneous, and log-concave function on the whole N\'eron--Severi
space, thus giving rise to a basic invariant of the underlying projective
variety. Analogously, one can also define the volume function of a possibly
non-complete multigraded linear series. In this paper we will address the
question of characterizing the class of functions arising on the one hand as
volume functions of multigraded linear series and on the other hand as volume
functions of projective varieties. In the multigraded setting, relying on the
work of Lazarsfeld and Musta\c{t}\u{a} (2009) on Okounkov bodies, we show that
any continuous, homogeneous, and log-concave function appears as the volume
function of a multigraded linear series. By contrast we show that there exists
countably many functions which arise as the volume functions of projective
varieties. We end the paper with an example, where the volume function of a
projective variety is given by a transcendental formula, emphasizing the
complicated nature of the volume in the classical case.Comment: 16 pages, minor revisio
On the number of tetrahedra with minimum, unit, and distinct volumes in three-space
We formulate and give partial answers to several combinatorial problems on
volumes of simplices determined by points in 3-space, and in general in
dimensions. (i) The number of tetrahedra of minimum (nonzero) volume spanned by
points in \RR^3 is at most , and there are point sets
for which this number is . We also present an time
algorithm for reporting all tetrahedra of minimum nonzero volume, and thereby
extend an algorithm of Edelsbrunner, O'Rourke, and Seidel. In general, for
every k,d\in \NN, , the maximum number of -dimensional
simplices of minimum (nonzero) volume spanned by points in \RR^d is
. (ii) The number of unit-volume tetrahedra determined by
points in \RR^3 is , and there are point sets for which this
number is . (iii) For every d\in \NN, the minimum
number of distinct volumes of all full-dimensional simplices determined by
points in \RR^d, not all on a hyperplane, is .Comment: 19 pages, 3 figures, a preliminary version has appeard in proceedings
of the ACM-SIAM Symposium on Discrete Algorithms, 200
Twitter-based analysis of the dynamics of collective attention to political parties
Large-scale data from social media have a significant potential to describe
complex phenomena in real world and to anticipate collective behaviors such as
information spreading and social trends. One specific case of study is
represented by the collective attention to the action of political parties. Not
surprisingly, researchers and stakeholders tried to correlate parties' presence
on social media with their performances in elections. Despite the many efforts,
results are still inconclusive since this kind of data is often very noisy and
significant signals could be covered by (largely unknown) statistical
fluctuations. In this paper we consider the number of tweets (tweet volume) of
a party as a proxy of collective attention to the party, identify the dynamics
of the volume, and show that this quantity has some information on the
elections outcome. We find that the distribution of the tweet volume for each
party follows a log-normal distribution with a positive autocorrelation of the
volume over short terms, which indicates the volume has large fluctuations of
the log-normal distribution yet with a short-term tendency. Furthermore, by
measuring the ratio of two consecutive daily tweet volumes, we find that the
evolution of the daily volume of a party can be described by means of a
geometric Brownian motion (i.e., the logarithm of the volume moves randomly
with a trend). Finally, we determine the optimal period of averaging tweet
volume for reducing fluctuations and extracting short-term tendencies. We
conclude that the tweet volume is a good indicator of parties' success in the
elections when considered over an optimal time window. Our study identifies the
statistical nature of collective attention to political issues and sheds light
on how to model the dynamics of collective attention in social media.Comment: 16 pages, 7 figures, 3 tables. Published in PLoS ON
Omega from the skewness of the cosmic velocity divergence
We propose a method for measuring the cosmological density parameter
from the statistics of the divergence field, , the
divergence of peculiar velocity, expressed in units of the Hubble constant, . The velocity field is spatially smoothed over to remove strongly nonlinear effects. Assuming weakly-nonlinear
gravitational evolution from Gaussian initial fluctuations, and using
second-order perturbative analysis, we show that \propto
-\Omega^{-0.6} ^2. The constant of proportionality depends on the
smoothing window. For a top-hat of radius R and volume-weighted smoothing, this
constant is , where . If the
power spectrum is a power law, , then . A Gaussian
window yields similar results. The resulting method for measuring is
independent of any assumed biasing relation between galaxies and mass.
The method has been successfully tested with numerical simulations. A
preliminary application to real data, provided by the POTENT recovery procedure
from observed velocities favors . However, because of an
uncertain sampling error, this result should be treated as an assessment of the
feasibility of our method rather than a definitive measurement of .Comment: 16 pages + 2 figures, uuencoded postscript file, also available by
anonymous ftp from ftp.cita.utoronto.ca in directory
/cita/francis/div_skewness, CITA 94-1
On the combinatorial classification of toric log del Pezzo surfaces
Toric log del Pezzo surfaces correspond to convex lattice polygons containing
the origin in their interior and having only primitive vertices. An upper bound
on the volume and on the number of boundary lattice points of these polygons is
derived in terms of the index l. Techniques for classifying these polygons are
also described: a direct classification for index two is given, and a
classification for all l<17 is obtained.Comment: 16 page
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