2,977 research outputs found
Reconstructing WKB from topological recursion
We prove that the topological recursion reconstructs the WKB expansion of a
quantum curve for all spectral curves whose Newton polygons have no interior
point (and that are smooth as affine curves). This includes nearly all
previously known cases in the literature, and many more; in particular, it
includes many quantum curves of order greater than two. We also explore the
connection between the choice of ordering in the quantization of the spectral
curve and the choice of integration divisor to reconstruct the WKB expansion.Comment: 68 pages, 9 figures. v2: published version (improved presentation
Convergence of algorithms for reconstructing convex bodies and directional measures
We investigate algorithms for reconstructing a convex body in from noisy measurements of its support function or its brightness
function in directions . The key idea of these algorithms is
to construct a convex polytope whose support function (or brightness
function) best approximates the given measurements in the directions
(in the least squares sense). The measurement errors are assumed
to be stochastically independent and Gaussian. It is shown that this procedure
is (strongly) consistent, meaning that, almost surely, tends to in
the Hausdorff metric as . Here some mild assumptions on the
sequence of directions are needed. Using results from the theory of
empirical processes, estimates of rates of convergence are derived, which are
first obtained in the metric and then transferred to the Hausdorff
metric. Along the way, a new estimate is obtained for the metric entropy of the
class of origin-symmetric zonoids contained in the unit ball. Similar results
are obtained for the convergence of an algorithm that reconstructs an
approximating measure to the directional measure of a stationary fiber process
from noisy measurements of its rose of intersections in directions
. Here the Dudley and Prohorov metrics are used. The methods are
linked to those employed for the support and brightness function algorithms via
the fact that the rose of intersections is the support function of a projection
body.Comment: Published at http://dx.doi.org/10.1214/009053606000000335 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Entwinement and the emergence of spacetime
It is conventional to study the entanglement between spatial regions of a
quantum field theory. However, in some systems entanglement can be dominated by
"internal", possibly gauged, degrees of freedom that are not spatially
organized, and that can give rise to gaps smaller than the inverse size of the
system. In a holographic context, such small gaps are associated to the
appearance of horizons and singularities in the dual spacetime. Here, we
propose a concept of entwinement, which is intended to capture this fine
structure of the wavefunction. Holographically, entwinement probes the
entanglement shadow -- the region of spacetime not probed by the minimal
surfaces that compute spatial entanglement in the dual field theory. We
consider the simplest example of this scenario -- a 2d conformal field theory
(CFT) that is dual to a conical defect in AdS3 space. Following our previous
work, we show that spatial entanglement in the CFT reproduces spacetime
geometry up to a finite distance from the conical defect. We then show that the
interior geometry up to the defect can be reconstructed from entwinement that
is sensitive to the discretely gauged, fractionated degrees of freedom of the
CFT. Entwinement in the CFT is related to non-minimal geodesics in the conical
defect geometry, suggesting a potential quantum information theoretic meaning
for these objects in a holographic context. These results may be relevant for
the reconstruction of black hole interiors from a dual field theory.Comment: v2: Sec. 4.3 amende
Turan's method and compressive sampling
Turan's method, as expressed in his books, is a careful study of
trigonometric polynomials from different points of view. The present article
starts from a problem asked by Turan: how to construct a sequence of real
numbers x(j) (j= 1,2,...n) such that the almost periodic polynomial whose
frequencies are the x(j) and the coefficients are 1 are small (say, their
absolute values are less than n d, d< given) for all integral values of the
variable m between 1 and M= M(n,d) ? The best known answer is a random choice
of the x(j) modulo 1. Using the random choice as Turan (and before him Erd\"os
and Renyi), we improve the estimate of M (n, d) and we discuss an explicit
construction derived from another chapter of Turan's book. The main part of the
paper deals with the corresponding problem when R / Z is replaced by Z / NZ, N
prime, and m takes all integral values modulo 1 except 0. Then it has an
interpretation in signal theory, when a signal is representad by a function on
the cyclic goup G = Z / NZ and the frequencies by the dual cyclic group G^ :
knowing that the signal is carried by T points, to evaluate the probability
that a random choice of a set W of frequencies allows to recover the signal x
from the restriction of its Fourier tranform to W by the process of minimal
extrapolation in the Wiener algebra of G^(process of Cand\`es, Romberg and
Tao). Some random choices were considered in the original article of CRT and
the corresponding probabilities were estimated in two preceding papers of mine.
Here we have another random choice, associated with occupancy problems
Tracing evolutionary links between species
The idea that all life on earth traces back to a common beginning dates back
at least to Charles Darwin's {\em Origin of Species}. Ever since, biologists
have tried to piece together parts of this `tree of life' based on what we can
observe today: fossils, and the evolutionary signal that is present in the
genomes and phenotypes of different organisms. Mathematics has played a key
role in helping transform genetic data into phylogenetic (evolutionary) trees
and networks. Here, I will explain some of the central concepts and basic
results in phylogenetics, which benefit from several branches of mathematics,
including combinatorics, probability and algebra.Comment: 18 pages, 6 figures (Invited review paper (draft version) for AMM
An Achievable Rate-Distortion Region for the Multiple Descriptions Problem
A multiple-descriptions (MD) coding strategy is proposed and an inner bound
to the achievable rate-distortion region is derived. The scheme utilizes linear
codes. It is shown in two different MD set-ups that the linear coding scheme
achieves a larger rate-distortion region than previously known random coding
strategies. Furthermore, it is shown via an example that the best known random
coding scheme for the set-up can be improved by including additional randomly
generated codebooks
DUDE-Seq: Fast, Flexible, and Robust Denoising for Targeted Amplicon Sequencing
We consider the correction of errors from nucleotide sequences produced by
next-generation targeted amplicon sequencing. The next-generation sequencing
(NGS) platforms can provide a great deal of sequencing data thanks to their
high throughput, but the associated error rates often tend to be high.
Denoising in high-throughput sequencing has thus become a crucial process for
boosting the reliability of downstream analyses. Our methodology, named
DUDE-Seq, is derived from a general setting of reconstructing finite-valued
source data corrupted by a discrete memoryless channel and effectively corrects
substitution and homopolymer indel errors, the two major types of sequencing
errors in most high-throughput targeted amplicon sequencing platforms. Our
experimental studies with real and simulated datasets suggest that the proposed
DUDE-Seq not only outperforms existing alternatives in terms of
error-correction capability and time efficiency, but also boosts the
reliability of downstream analyses. Further, the flexibility of DUDE-Seq
enables its robust application to different sequencing platforms and analysis
pipelines by simple updates of the noise model. DUDE-Seq is available at
http://data.snu.ac.kr/pub/dude-seq
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