7,993 research outputs found
A short proof of a near-optimal cardinality estimate for the product of a sum set
In this note it is established that, for any finite set of real numbers,
there exist two elements such that
In particular, it follows that . The
latter inequality had in fact already been established in an earlier work of
the author and Rudnev (arXiv:1203.6237), which built upon the recent
developments of Guth and Katz (arXiv:1011.4105) in their work on the Erd\H{o}s
distinct distance problem. Here, we do not use those relatively deep methods,
and instead we need just a single application of the Szemer\'{e}di-Trotter
Theorem. The result is also qualitatively stronger than the corresponding
sum-product estimate from (arXiv:1203.6237), since the set is
defined by only two variables, rather than four. One can view this as a
solution for the pinned distance problem, under an alternative notion of
distance, in the special case when the point set is a direct product . Another advantage of this more elementary approach is that these results
can now be extended for the first time to the case when .Comment: To appear in Proceedings of SoCG 201
Local energy statistics in disordered systems: a proof of the local REM conjecture
Recently, Bauke and Mertens conjectured that the local statistics of energies
in random spin systems with discrete spin space should in most circumstances be
the same as in the random energy model. Here we give necessary conditions for
this hypothesis to be true, which we show to hold in wide classes of examples:
short range spin glasses and mean field spin glasses of the SK type. We also
show that, under certain conditions, the conjecture holds even if energy levels
that grow moderately with the volume of the system are considered
The Littlewood-Offord Problem and invertibility of random matrices
We prove two basic conjectures on the distribution of the smallest singular
value of random n times n matrices with independent entries. Under minimal
moment assumptions, we show that the smallest singular value is of order
n^{-1/2}, which is optimal for Gaussian matrices. Moreover, we give a optimal
estimate on the tail probability. This comes as a consequence of a new and
essentially sharp estimate in the Littlewood-Offord problem: for i.i.d. random
variables X_k and real numbers a_k, determine the probability P that the sum of
a_k X_k lies near some number v. For arbitrary coefficients a_k of the same
order of magnitude, we show that they essentially lie in an arithmetic
progression of length 1/p.Comment: Introduction restructured, some typos and minor errors correcte
On the linear independence of spikes and sines
The purpose of this work is to survey what is known about the linear
independence of spikes and sines. The paper provides new results for the case
where the locations of the spikes and the frequencies of the sines are chosen
at random. This problem is equivalent to studying the spectral norm of a random
submatrix drawn from the discrete Fourier transform matrix. The proof involves
depends on an extrapolation argument of Bourgain and Tzafriri.Comment: 16 pages, 4 figures. Revision with new proof of major theorem
Compressive sensing Petrov-Galerkin approximation of high-dimensional parametric operator equations
We analyze the convergence of compressive sensing based sampling techniques
for the efficient evaluation of functionals of solutions for a class of
high-dimensional, affine-parametric, linear operator equations which depend on
possibly infinitely many parameters. The proposed algorithms are based on
so-called "non-intrusive" sampling of the high-dimensional parameter space,
reminiscent of Monte-Carlo sampling. In contrast to Monte-Carlo, however, a
functional of the parametric solution is then computed via compressive sensing
methods from samples of functionals of the solution. A key ingredient in our
analysis of independent interest consists in a generalization of recent results
on the approximate sparsity of generalized polynomial chaos representations
(gpc) of the parametric solution families, in terms of the gpc series with
respect to tensorized Chebyshev polynomials. In particular, we establish
sufficient conditions on the parametric inputs to the parametric operator
equation such that the Chebyshev coefficients of the gpc expansion are
contained in certain weighted -spaces for . Based on this we
show that reconstructions of the parametric solutions computed from the sampled
problems converge, with high probability, at the , resp.
convergence rates afforded by best -term approximations of the parametric
solution up to logarithmic factors.Comment: revised version, 27 page
A Numerical Approach for Designing Unitary Space Time Codes with Large Diversity
A numerical approach to design unitary constellation for any dimension and
any transmission rate under non-coherent Rayleigh flat fading channel.Comment: 32 pages, 6 figure
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