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 $S^2$. 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 $\mathbb R^n$, and improve bounds on the number $n(d,k)$ in the analogous conjecture for odd degrees $d$ (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 $d=2$ (for $k$'s of certain type), odd $d$, and the complex Grassmannian (for odd and even $d$ and any $k$). Corollaries for the John ellipsoid of projections or sections of a convex body are deduced from the case $d=2$ 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

Knaster's problem for (Z2)k(Z_2)^k-symmetric subsets of the sphere S2k−1S^{2^k-1}

We prove a Knaster-type result for orbits of the group $(Z_2)^k$ in $S^{2^k-1}$, calculating the Euler class obstruction. Among the consequences are: a result about inscribing skew crosspolytopes in hypersurfaces in $\mathbb R^{2^k}$, and a result about equipartition of a measures in $\mathbb R^{2^k}$ by $(Z_2)^{k+1}$-symmetric convex fans

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

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