292 research outputs found
Almost Euclidean sections of the N-dimensional cross-polytope using O(N) random bits
It is well known that R^N has subspaces of dimension proportional to N on
which the \ell_1 norm is equivalent to the \ell_2 norm; however, no explicit
constructions are known. Extending earlier work by Artstein--Avidan and Milman,
we prove that such a subspace can be generated using O(N) random bits.Comment: 16 pages; minor changes in the introduction to make it more
accessible to both Math and CS reader
Expected Supremum of a Random Linear Combination of Shifted Kernels
We address the expected supremum of a linear combination of shifts of the
sinc kernel with random coefficients. When the coefficients are Gaussian, the
expected supremum is of order \sqrt{\log n}, where n is the number of shifts.
When the coefficients are uniformly bounded, the expected supremum is of order
\log\log n. This is a noteworthy difference to orthonormal functions on the
unit interval, where the expected supremum is of order \sqrt{n\log n} for all
reasonable coefficient statistics.Comment: To appear in the Journal of Fourier Analysis and Application
A note on Makeev's conjectures
A counterexample is given for the Knaster-like conjecture of Makeev for
functions on . Some particular cases of another conjecture of Makeev, on
inscribing a quadrangle into a smooth simple closed curve, are solved
positively
Dvoretzky type theorems for multivariate polynomials and sections of convex bodies
In this paper we prove the Gromov--Milman conjecture (the Dvoretzky type
theorem) for homogeneous polynomials on , and improve bounds on
the number in the analogous conjecture for odd degrees (this case
is known as the Birch theorem) and complex polynomials. We also consider a
stronger conjecture on the homogeneous polynomial fields in the canonical
bundle over real and complex Grassmannians. This conjecture is much stronger
and false in general, but it is proved in the cases of (for 's of
certain type), odd , and the complex Grassmannian (for odd and even and
any ). Corollaries for the John ellipsoid of projections or sections of a
convex body are deduced from the case of the polynomial field conjecture
From Low-Distortion Norm Embeddings to Explicit Uncertainty Relations and Efficient Information Locking
The existence of quantum uncertainty relations is the essential reason that
some classically impossible cryptographic primitives become possible when
quantum communication is allowed. One direct operational manifestation of these
uncertainty relations is a purely quantum effect referred to as information
locking. A locking scheme can be viewed as a cryptographic protocol in which a
uniformly random n-bit message is encoded in a quantum system using a classical
key of size much smaller than n. Without the key, no measurement of this
quantum state can extract more than a negligible amount of information about
the message, in which case the message is said to be "locked". Furthermore,
knowing the key, it is possible to recover, that is "unlock", the message. In
this paper, we make the following contributions by exploiting a connection
between uncertainty relations and low-distortion embeddings of L2 into L1. We
introduce the notion of metric uncertainty relations and connect it to
low-distortion embeddings of L2 into L1. A metric uncertainty relation also
implies an entropic uncertainty relation. We prove that random bases satisfy
uncertainty relations with a stronger definition and better parameters than
previously known. Our proof is also considerably simpler than earlier proofs.
We apply this result to show the existence of locking schemes with key size
independent of the message length. We give efficient constructions of metric
uncertainty relations. The bases defining these metric uncertainty relations
are computable by quantum circuits of almost linear size. This leads to the
first explicit construction of a strong information locking scheme. Moreover,
we present a locking scheme that is close to being implementable with current
technology. We apply our metric uncertainty relations to exhibit communication
protocols that perform quantum equality testing.Comment: 60 pages, 5 figures. v4: published versio
Is there a common origin for the WMAP low multipole and for the ellipticity in BOOMERanG CMB maps?
We have measured the ellipticity of several degree scale anisotropies in the
BOOMERanG maps of the Cosmic Microwave Background (CMB) at 150 GHz. The average
ellipticity is around 2.6-2.7. The biases of the estimator of the ellipticity
and for the noise are small in this case. Large spot elongation had been
detected also for COBE-DMR maps. If this effect is due to geodesic mixing, it
would indicate a non precisely zero curvature of the Universe which is among
the discussed reasons of the WMAP low multipole anomaly. Both effects are
related to the diameter of the Universe: the geodesics mixing through
hyperbolic geometry, low multipoles through boundary conditions.This common
reason can also be related with the origin of the the cosmological constant:
the modes of vacuum fluctuations conditioned by the boundary conditions lead to
a value of the cosmological constant being in remarkable agreement with the
supernovae observations.Comment: Added: two co-authors and a comment on the possible relation of the
discussed CMB properties with the origin of the observed value of the
cosmological constan
Knaster's problem for -symmetric subsets of the sphere
We prove a Knaster-type result for orbits of the group in
, calculating the Euler class obstruction. Among the consequences
are: a result about inscribing skew crosspolytopes in hypersurfaces in , and a result about equipartition of a measures in
by -symmetric convex fans
WMAP confirming the ellipticity in BOOMERanG and COBE CMB maps
The recent study of BOOMERanG 150 GHz Cosmic Microwave Background (CMB)
radiation maps have detected ellipticity of the temperature anisotropy spots
independent on the temperature threshold. The effect has been found for spots
up to several degrees in size, where the biases of the ellipticity estimator
and of the noise are small. To check the effect, now we have studied, with the
same algorithm and in the same sky region, the WMAP maps. We find ellipticity
of the same average value also in WMAP maps, despite of the different
sensitivity of the two experiments to low multipoles. Large spot elongations
had been detected also for the COBE-DMR maps. If this effect is due to geodesic
mixing and hence due to non precisely zero curvature of the hyperbolic
Universe, it can be linked to the origin of WMAP low multipoles anomaly.Comment: More explanations and two references adde
Detection of X-ray galaxy clusters based on the Kolmogorov method
The detection of clusters of galaxies in large surveys plays an important
part in extragalactic astronomy, and particularly in cosmology, since cluster
counts can give strong constraints on cosmological parameters. X-ray imaging is
in particular a reliable means to discover new clusters, and large X-ray
surveys are now available. Considering XMM-Newton data for a sample of 40 Abell
clusters, we show that their analysis with a Kolmogorov distribution can
provide a distinctive signature for galaxy clusters. The Kolmogorov method is
sensitive to the correlations in the cluster X-ray properties and can therefore
be used for their identification, thus allowing to search reliably for clusters
in a simple way
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