45,562 research outputs found
Counting projections of rational curves
Given two general rational curves of the same degree in two projective
spaces, one can ask whether there exists a third rational curve of the same
degree that projects to both of them. We show that, under suitable assumptions
on the degree of the curves and the dimensions of the two given ambient
projective spaces, the number of curves and projections fulfilling the
requirements is finite. Using standard techniques in intersection theory and
the Bott residue formula, we compute this number.Comment: 27 page
The Fractal Density Structure in Supersonic Isothermal Turbulence: Solenoidal versus Compressive Energy Injection
In a systematic study, we compare the density statistics in high resolution
numerical experiments of supersonic isothermal turbulence, driven by the
usually adopted solenoidal (divergence-free) forcing and by compressive
(curl-free) forcing. We find that for the same rms Mach number, compressive
forcing produces much stronger density enhancements and larger voids compared
to solenoidal forcing. Consequently, the Fourier spectra of density
fluctuations are significantly steeper. This result is confirmed using the
Delta-variance analysis, which yields power-law exponents beta~3.4 for
compressive forcing and beta~2.8 for solenoidal forcing. We obtain fractal
dimension estimates from the density spectra and Delta-variance scaling, and by
using the box counting, mass size and perimeter area methods applied to the
volumetric data, projections and slices of our turbulent density fields. Our
results suggest that compressive forcing yields fractal dimensions
significantly smaller compared to solenoidal forcing. However, the actual
values depend sensitively on the adopted method, with the most reliable
estimates based on the Delta-variance, or equivalently, on Fourier spectra.
Using these methods, we obtain D~2.3 for compressive and D~2.6 for solenoidal
forcing, which is within the range of fractal dimension estimates inferred from
observations (D~2.0-2.7). The velocity dispersion to size relations for both
solenoidal and compressive forcing obtained from velocity spectra follow a
power law with exponents in the range 0.4-0.5, in good agreement with previous
studies.Comment: 17 pages, 11 figures, ApJ in press, minor changes to language,
simulation movies available at
http://www.ita.uni-heidelberg.de/~chfeder/videos.shtml?lang=e
Estimating Entropy of Data Streams Using Compressed Counting
The Shannon entropy is a widely used summary statistic, for example, network
traffic measurement, anomaly detection, neural computations, spike trains, etc.
This study focuses on estimating Shannon entropy of data streams. It is known
that Shannon entropy can be approximated by Reenyi entropy or Tsallis entropy,
which are both functions of the p-th frequency moments and approach Shannon
entropy as p->1.
Compressed Counting (CC) is a new method for approximating the p-th frequency
moments of data streams. Our contributions include:
1) We prove that Renyi entropy is (much) better than Tsallis entropy for
approximating Shannon entropy.
2) We propose the optimal quantile estimator for CC, which considerably
improves the previous estimators.
3) Our experiments demonstrate that CC is indeed highly effective
approximating the moments and entropies. We also demonstrate the crucial
importance of utilizing the variance-bias trade-off
Sets which are not tube null and intersection properties of random measures
We show that in there are purely unrectifiable sets of
Hausdorff (and even box counting) dimension which are not tube null,
settling a question of Carbery, Soria and Vargas, and improving a number of
results by the same authors and by Carbery. Our method extends also to "convex
tube null sets", establishing a contrast with a theorem of Alberti,
Cs\"{o}rnyei and Preiss on Lipschitz-null sets. The sets we construct are
random, and the proofs depend on intersection properties of certain random
fractal measures with curves.Comment: 24 pages. Referees comments incorporated. JLMS to appea
On Practical Algorithms for Entropy Estimation and the Improved Sample Complexity of Compressed Counting
Estimating the p-th frequency moment of data stream is a very heavily studied
problem. The problem is actually trivial when p = 1, assuming the strict
Turnstile model. The sample complexity of our proposed algorithm is essentially
O(1) near p=1. This is a very large improvement over the previously believed
O(1/eps^2) bound. The proposed algorithm makes the long-standing problem of
entropy estimation an easy task, as verified by the experiments included in the
appendix
Counting faces of randomly-projected polytopes when the projection radically lowers dimension
This paper develops asymptotic methods to count faces of random
high-dimensional polytopes. Beyond its intrinsic interest, our conclusions have
surprising implications - in statistics, probability, information theory, and
signal processing - with potential impacts in practical subjects like medical
imaging and digital communications. Three such implications concern: convex
hulls of Gaussian point clouds, signal recovery from random projections, and
how many gross errors can be efficiently corrected from Gaussian error
correcting codes.Comment: 56 page
Counting higher genus curves in a Calabi-Yau manifold
We explicitly evaluate the low energy coupling in a
compactification of the heterotic string. The holomorphic piece of this
expression provides the information not encoded in the holomorphic anomaly
equations, and we find that it is given by an elementary polylogarithm with
index , thus generalizing in a natural way the known results for .
The heterotic model has a dual Calabi-Yau compactification of the type II
string. We compare the answer with the general form expected from
curve-counting formulae and find good agreement. As a corollary of this
comparison we predict some numbers of higher genus curves in a specific
Calabi-Yau, and extract some intersection numbers on the moduli space of genus
Riemann surfaces.Comment: 26 pages, harvmac b mod
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